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Prove that cos^2(A/2)+cos^2(B/2)+cos^2(C/2)=2.cos(A/2)cos(B/2).sin(C/2)

2021-09-09T18:24:00.001+05:30

Steps to prove cos^2(A/2)+cos^2(B/2)+cos^2(C/2)=2.cos(A/2)cos(B/2).sin(C/2)

L.H.S

=cos^2(A/2)+cos^2(B/2)+cos^2(C/2)

=(1+cosA)/2+(1+cosB)/2-(1+cosC)/2

=(1+cosA+1+cosB-1-cosC)/2

=(1+cosA+cosB-cosC)/2

=1/2+(cosA+cosB-cosC)/2

=1/2+1/2{2.sin(C/2).cos(A-B)/2-1+2sin^2(C/2)}

=1/2+1/2[2.sin(C/2).{cos(A-B)/2+sin(C/2)}-1]

=1/2+1/2[2.sin(C/2).2cos(A/2).cos(B/2)-1]

=1/2+2.cos(A/2)cos(B/2).sin(C/2)-1/2

=2.cos(A/2)cos(B/2).sin(C/2)

R.H.S

Hence cos^2(A/2)+cos^2(B/2)+cos^2(C/2)=2.cos(A/2)cos(B/2).sin(C/2) is proved.

Detail Information:-

Prove that cos^3A.cos3A+sin^3A.sin3A=cos^32A
Prove that cos^6A+sin^6A=(1/4)(1+3cos^2(2A))
Prove that sin^4θ=3/8-(1/2)cos2θ+(1/8)cos4θ
Prove that cot3A=(cot^3A-3cotA)/(3cot^2A-1)

If x+y+z=xyz then prove that x/(1-x^2)+y/(1-y^2)+z/(1-z^2)=xyz/{(1-x^2)(1-y^2)(1-z^2)}

2021-09-09T18:17:00.001+05:30

Steps to prove x/(1-x^2)+y/(1-y^2)+z/(1-z^2)=xyz/{(1-x^2)(1-y^2)(1-z^2)}

Let us consider

x=tanα, y=tanβ, z=tanγ

⇒x+y+z=xyz

⇒tanα+tanβ+tanγ=tanα.tanβ.tanγ

⇒tanα+tanβ=tanα.tanβ.tanγ-tanγ

⇒tanα+tanβ=tanγ(tanα.tanβ-1)

⇒(tanα+tanβ)/(1-tanα.tanβ)=-tanγ

⇒tan(α+β)=-tanγ

⇒tan(α+β)=tan(pi-γ)

⇒α+β=pi-γ

⇒2α+2β=2pi-2γ

⇒tan(2α+2β)=tan(2pi-2γ)

⇒(tan2α+tan2β)/(1-tan2α.tan2β)=-tan2γ

⇒tan2α+tan2β+tan2γ=tan2α.tan2β.tan2γ

⇒2x/(1-x^2)+2y/(1-y^2)+2z/(1-z^2)=2x/(1-x^2).2y/(1-y^2).2z/(1-z^2)

⇒x/(1-x^2)+y/(1-y^2)+z/(1-z^2)=xyz/{(1-x^2)(1-y^2)(1-z^2)}

Hence x/(1-x^2)+y/(1-y^2)+z/(1-z^2)=xyz/{(1-x^2)(1-y^2)(1-z^2)} is proved.

Detail Information:-

An angle θ is divide into two parts α,β such that tanα:tanβ=x:y then prove that sin(α-β)={(x-y)/(x+y)}sinθ
If (1-e)tan^2(β/2)=(1+e)tan^2(α/2), then prove that cosβ=(cosα-e)/(1-ecosα)
If tanA+tanB=a and cotA+cotB=b, then prove that cot(A+B)=1/a-1/b
If sec(φ+α)+sec(φ-α)=2secφ show that cosφ=+-√cosα/2

Prove that sin^2(A/2)+sin^2(B/2)+sin^2(C/2)=1-2.sin(A/2)sin(B/2).sin(C/2)

2021-09-09T18:11:00.000+05:30

Steps to prove sin^2(A/2)+sin^2(B/2)+sin^2(C/2)=1-2.sin(A/2)sin(B/2).sin(C/2)

L.H.S

=sin^2(A/2)+sin^2(B/2)+sin^2(C/2)

=(1-cosA)/2+(1-cosB)/2+(1-cosC)/2

=(1-cosA+1-cosB+1-cosC)/2

={3-(cosA+cosB+cosC)}/2

=(1/2)[3-{2.cos(A+B)/2.cos(A-B)/2+1-2sin^2(c/2)}]

=(1/2)[3-2.cos(A+B)/2.cos(A-B)/2-1+2sin^2(c/2)}]

=(1/2)[2-2sin(C/2){cos(A-B)/2-cos(A+B)/2}]

=(1/2).2[1-sin(C/2){cos(A-B)/2-cos(A+B)/2}]

=1-sin(C/2){cos(A/2-B/2)-cos(A/2+B/2)}

=1-2.sin(A/2)sin(B/2).sin(C/2)

Hence sin^2(A/2)+sin^2(B/2)+sin^2(C/2)=1-2.sin(A/2)sin(B/2).sin(C/2) is proved.

Detail Information:-

An angle θ is divide into two parts α,β such that tanα:tanβ=x:y then prove that sin(α-β)={(x-y)/(x+y)}sinθ
If (1-e)tan^2(β/2)=(1+e)tan^2(α/2), then prove that cosβ=(cosα-e)/(1-ecosα)
If tanA+tanB=a and cotA+cotB=b, then prove that cot(A+B)=1/a-1/b
If sec(φ+α)+sec(φ-α)=2secφ show that cosφ=+-√cosα/2

An angle θ is divide into two parts α,β such that tanα:tanβ=x:y then prove that sin(α-β)={(x-y)/(x+y)}sinθ

2021-09-08T12:32:00.002+05:30

Steps to prove sin(α-β)={(x-y)/(x+y)}sinθ

From Question We know that

⇒tanα:tanβ=x:y

⇒tanα/tanβ=x/y

⇒(sinα/cosα)/(sinβ/cosβ)=x/y

⇒(sinα.cosβ)/(cosβ.sinα)=x/y

By componendo and dividendo

⇒{(sinα.cosβ)+(cosα.sinβ)}/{(cosβ.sinα)-(cosα.sinβ)}=(x+y)/(x-y)

⇒sin(α+β)/sin(α-β)=(x+y)/(x-y)

⇒sinθ/sin(α-β)=(x+y)/(x-y)

⇒sin(α-β)={(x-y)/(x+y)}sinθ

Hence sin(α-β)={(x-y)/(x+y)}sinθ is proved.

Detail Information:-

If (1-e)tan^2(β/2)=(1+e)tan^2(α/2), then prove that cosβ=(cosα-e)/(1-ecosα)
If tanA+tanB=a and cotA+cotB=b, then prove that cot(A+B)=1/a-1/b
If sec(φ+α)+sec(φ-α)=2secφ show that cosφ=+-√cosα/2
If A+B+C= π than prove that sin2A+sin2B-sin2C=4.cosA.cosB.sinC

If (1-e)tan^2(β/2)=(1+e)tan^2(α/2), then prove that cosβ=(cosα-e)/(1-ecosα)

2021-09-08T12:25:00.000+05:30

Steps to prove cosβ=(cosα-e)/(1-ecosα)

From Question

⇒(1-e)tan^2(β/2)=(1+e)tan^2(α/2)

⇒tan^2(β/2)={(1+e)/(1-e)}tan^2(α/2)

L.H.S

=cosβ

={1-tan^2(β/2)}/{1+tan^2(β/2)}

=[1-{(1+e)/(1-e)}tan^2(α/2)]/[1+{(1+e)/(1-e)}tan^2(α/2)]

=[(1-e)-{tan^2(α/2)+etan^2(α/2)}]/[(1-e)+{tan^2(α/2)+etan^2(α/2)}]

={1-tan^2(α/2)-e-etan^2(α/2)}/{1+tan^2(α/2)-e+etan^2(α/2)

Dividing 1+tan^2(α/2) both numenetor and denomenator

=[{1-tan^2(α/2)}/{1+tan^2(α/2)}-e{1+tan^2(α/2)}/{1+tan^2(α/2)}]/[{1+tan^2(α/2)}/{1+tan^2(α/2)}-e{1-tan^2(α/2)}/{1+tan^2(α/2)}]

=(cosα-e)/(1-ecosα)

=R.H.S

Hence cosβ=(cosα-e)/(1-ecosα) is proved.

Detail Information:-

If tanA+tanB=a and cotA+cotB=b, then prove that cot(A+B)=1/a-1/b
If sec(φ+α)+sec(φ-α)=2secφ show that cosφ=+-√cosα/2
If A+B+C= π than prove that sin2A+sin2B-sin2C=4.cosA.cosB.sinC
If A+B+C= π than prove that cosA+cosB+cosC=1+4.sin(1/2)A.sin(1/2)B.sin(1/2)C

If tanA+tanB=a and cotA+cotB=b, then prove that cot(A+B)=1/a-1/b

2021-09-08T12:18:00.000+05:30

Steps to prove cot(A+B)=1/a-1/b

R.H.S

=1/a-1/b

=1/(tanA+tanB)-1/(cotA+cotB)

=1/(1/cotA+1/cotB)-1/(cotA+cotB)

=cotA.cotB/(cotB+cotA)-1/(cotA+cotB)

=(cotA.cotB-1)/(cotB+cotA)

=cot(A+B)

L.H.S

Hence cot(A+B)=1/a-1/b is proved.

Detail Information:-

If sec(φ+α)+sec(φ-α)=2secφ show that cosφ=+-√cosα/2
If A+B+C= π than prove that sin2A+sin2B-sin2C=4.cosA.cosB.sinC
If A+B+C= π than prove that sinA+sinB-sinC=4.sin(1/2)A.sin(1/2)B.cos(1/2)C
If A+B+C= π than prove that cosA+cosB+cosC=1+4.sin(1/2)A.sin(1/2)B.sin(1/2)C

If sec(φ+α)+sec(φ-α)=2secφ show that cosφ=+-√cosα/2

2021-09-08T09:39:00.004+05:30

Steps to prove cosφ=+-√cosα/2

From Question

⇒sec(φ+α)+sec(φ-α)=2secφ

⇒1/cos(φ+α)+1/cos(φ-α)=2/cosφ

⇒{cos(φ-α)+cos(φ+α)}/{cos(φ-α).cos(φ+α)}=2/cosφ

⇒2.cosφ.cosα/(cos^2φ-sin^2α)=2/cosφ

⇒cos^2φ.cosα=cos^2φ-sin^2α

⇒cos^2φ.cosα-cos^2φ=-sin^2α

⇒cos^2φ.cosα-cos^2φ=-(1-cos^2α)

⇒cos^2φ(cosα-1)=(cosα+1)(cosα-1)

⇒cos^2φ=1+cosα

⇒cos^2φ=2cos^2α/2

⇒cosφ=+-√cosα/2

Hence cosφ=+-√cosα/2 is proved

Detail Information:-

If α,β are acute angles and cos2α=(3cos2β-1)/(3-cos2β) than prove that tanα=√2tanβ
If 2tanα=3tanβ than prove that tanβ/(2+3tan^2β)
If acos(x+α)=bcos(x-α) show that (a+b)tanx=(a-b)cotα
Prove that cos7α+cos3α-cos5α-cosα/sin7α-sin3α-sin5α+sinα=cot2α

If A+B+C= π than prove that sin2A+sin2B-sin2C=4.cosA.cosB.sinC

2021-09-08T09:30:00.004+05:30

Steps to prove sin2A+sin2B-sin2C=4.cosA.cosB.sinC

L.H.S

=sin2A+sin2B-sin2C

=2sin(2A+2B)/2.cos(2A-2B)/2-2.sinC.cosC

=2sin(A+B).cos(A-B)-2.sinC.cosC

=2sinC{cos(A-B)-cosC} [sin(A+B)=sinC]

=2sinC{cos(A-B)+cos(A+B)} [cos(A+B)=-cosC]

=2.sinC.2.cosA.cosB

=4.cosA.cosB.cosC

=R.H.S

Hence sin2A+sin2B-sin2C=4.cosA.cosB.sinC is proved.

Detail Information:-

If A+B+C= π than prove that cosA+cosB+cosC=1+4.sin(1/2)A.sin(1/2)B.sin(1/2)C
If A+B+C= π than prove that sinA+sinB-sinC=4.sin(1/2)A.sin(1/2)B.cos(1/2)C
If α,β are acute angles and cos2α=(3cos2β-1)/(3-cos2β) than prove that tanα=√2tanβ
If cotθ=cos(x+y) and cotΦ=cos(x-y) show that tan(θ-Φ)=2.sinx.siny/(cos^2x+cos^2y)

If A+B+C= π than prove that cosA+cosB+cosC=1+4.sin(1/2)A.sin(1/2)B.sin(1/2)C

2021-09-08T09:22:00.004+05:30

Steps to prove cosA+cosB+cosC=1+4.sin(1/2)A.sin(1/2)B.sin(1/2)C

L.H.S

=cosA+cosB+cosC

=2.cos(A+B)/2.cos(A-B)/2+(1-2sin^2C/2)

=2.sinC/2.cos(A-B)/2+1-2sin^2C/2 [sinC/2=cos(A+B)/2]

=2sinC/2{cos(A/2-B/2)-cos(A/2+B/2)}+1

=2sinC/2.2sinA/2.sinB/2+1

=1+4.sin(1/2)A.sin(1/2)B.sin(1/2)C

=R.H.S

Hence cosA+cosB+cosC=1+4.sin(1/2)A.sin(1/2)B.sin(1/2)C is proved.

Detail Information:-

If A+B+C= π than prove that sinA+sinB-sinC=4.sin(1/2)A.sin(1/2)B.cos(1/2)C
If A+B+C=π that prove that tan(B+C)/2=cotA/2
If cotθ=cos(x+y) and cotΦ=cos(x-y) show that tan(θ-Φ)=2.sinx.siny/(cos^2x+cos^2y)
If acos(x+α)=bcos(x-α) show that (a+b)tanx=(a-b)cotα

If A+B+C= π than prove that sinA+sinB-sinC=4.sin(1/2)A.sin(1/2)B.cos(1/2)C

2021-09-08T09:13:00.001+05:30

Steps to prove sinA+sinB-sinC=4.sin(1/2)A.sin(1/2)B.cos(1/2)C

L.H.S

=sinA+sinB-sinC

=2sin(A+B)/2.cos(A-B)/2-2sinC/2.cosC/2

=2cosC/2.cos(A-B)/2-2sinC/2.cosC/2 [sin(A+B)/2=cosC/2]

=2cosC/2{cos(A-B)/2-sinC/2}

=2cosC/2{cos(A-B)/2-cos(A+B)/2} [sinC/2=cos(A+B)/2]

=2cosC/2.2.sinA/2.cosA/2

=4.sin(1/2)A.sin(1/2)B.cos(1/2)C

=R.H.S

Hence sinA+sinB-sinC=4.sin(1/2)A.sin(1/2)B.cos(1/2)C is proved.

Detail Information:-

If A+B+C=π that prove that tan(B+C)/2=cotA/2
If α,β are acute angles and cos2α=(3cos2β-1)/(3-cos2β) than prove that tanα=√2tanβ
If cotθ=cos(x+y) and cotΦ=cos(x-y) show that tan(θ-Φ)=2.sinx.siny/(cos^2x+cos^2y)
If acos(x+α)=bcos(x-α) show that (a+b)tanx=(a-b)cotα

If α,β are acute angles and cos2α=(3cos2β-1)/(3-cos2β) than prove that tanα=√2tanβ

2021-09-07T22:28:00.002+05:30

Steps to prove tanα=√2tanβ

We know that

⇒cos2α=(3cos2β-1)/(3-cos2β)

⇒(1-tan^2α)/(1+tan^2α)={3-(1-tan^2β)/(1+tan^2β)-1}/{3-(1-tan^2β)/(1+tan^2β)}

⇒(1-tan^2α)/(1+tan^2α)=(2-4tan^2β)/(2+4tan^2β)

⇒(1-tan^2α)/(1+tan^2α)=(1-2tan^2β)/(1+2tan^2β)

⇒{(1-tan^2α)+(1+tan^2α)}/{(1-tan^2α)-(1+tan^2α)}={(1-2tan^2β)+(1+2tan^2β)}/{(1-2tan^2β)+(1+2tan^2β)}

⇒2/(-2tan^2α)=2/(-4tan^2β)

⇒1/(tan^2α)=1/(2tan^2β)

⇒tan^2α=rt2tan^2β

⇒L.H.S=R.H.S

Hence tanα=√2tanβ is proved.

Detail Information:-

If 2tanα=3tanβ than prove that tanβ/(2+3tan^2β)
If cotθ=cos(x+y) and cotΦ=cos(x-y) show that tan(θ-Φ)=2.sinx.siny/(cos^2x+cos^2y)
Prove that tan37(1/2)=√6+√3-√2-2
Prove that cos(π/16)=(1/2)√2+√2+√2

If 2tanα=3tanβ than prove that tanβ/(2+3tan^2β)

2021-09-07T22:10:00.003+05:30

Steps to prove tan(α-β)=sin2β/(5-cos2β)

L.H.S

=tan(α-β)

=(tanα-tanβ)/(1+tanα.tanβ)

={(3/2)tanβ-tanβ}/{1+(3/2)tanβ.tanβ}

={(3tanβ-2tanβ)/2}/(2+3tan^2β)

=tanβ/(2+3tan^2β)

R.H.S

=sin2β/(5-cos2β)

={2tanβ/(1+tan^2β)}/{5-(1-tan^2β)/(1+tan^2β)}

={2tanβ/(1+tan^2β)}/[{5(1+tan^2β)-(1-tan^2β)}/(1+tan^2β)

=2tanβ/(4+6tan^2β)

=tanβ/(2+3tan^2β)

Hence L.H.S=R.H.S

tanβ/(2+3tan^2β) is proved.

Detail Information:-

If cotθ=cos(x+y) and cotΦ=cos(x-y) show that tan(θ-Φ)=2.sinx.siny/(cos^2x+cos^2y)
Prove that tan37(1/2)=√6+√3-√2-2
If sinA=ksinB, Prove that tan(1/2)(A-B)=(k-1)/(k+1)tan(1/2)(A+B)
If acos(x+α)=bcos(x-α) show that (a+b)tanx=(a-b)cotα

If cotθ=cos(x+y) and cotΦ=cos(x-y) show that tan(θ-Φ)=2.sinx.siny/(cos^2x+cos^2y)

2021-09-07T18:23:00.003+05:30

Steps to prove tan(θ-Φ)=2.sinx.siny/(cos^2x+cos^2y)

L.H.S

=tan(θ-Φ)

=(tanθ-tanΦ)/(1-tanθ.tanΦ)

=(1/cotθ-1/cotΦ)/(1-1/cotθ.1/cotΦ)

=(cotΦ-cotθ)/(cotθ.cotΦ+1)

={cos(x-y)-cos(x+y)}/{cos(x-y).cos(x+y)+1}

=2sinx.siny/(cos^2x-sin^2y+1)

=2sinx.siny/(cos^2x+1-sin^2y)

=2.sinx.siny/(cos^2x+cos^2y)

R.H.S

Hence tan(θ-Φ)=2.sinx.siny/(cos^2x+cos^2y) is proved.

Detail Information:-

Prove that tan37(1/2)=√6+√3-√2-2
Prove that cos(π/16)=(1/2)√2+√2+√2
If sinA=ksinB, Prove that tan(1/2)(A-B)=(k-1)/(k+1)tan(1/2)(A+B)
If acos(x+α)=bcos(x-α) show that (a+b)tanx=(a-b)cotα

Prove that tan37(1/2)=√6+√3-√2-2

2021-09-05T12:23:00.001+05:30

Steps to prove tan37(1/2)=√6+√3-√2-2

L.H.S

=tan37(1/2)

=tan(75/2)

=(1-cos75)/sin75

={1-(√3-1/2√2)}/(√3+1/2√2)

=(2√2-√3+1)/(√3-1)

=(2√2-√3+1)(√3-1)/(√3+1)(√3-1)

=(2√6+2√3-2√2-4)/3-1

=(2√6+2√3-2√2-4)/2

=2(√6+√3-√2-2)/2

=√6+√3-√2-2

=R.H.S

Hence tan37(1/2)=√6+√3-√2-2 is proved.

Detail Information:-

Prove that cot7(1/2)=√6+√3+√2+2
Prove that cot22(1/2)=√2+1
Prove that cot37(1/2)=√6-√3-√2+2
Prove that tanA/2=√(1-cosA)/(1+cosA)

Prove that cos(π/16)=(1/2)√2+√2+√2

2021-09-05T12:14:00.002+05:30

Steps to prove cos(π/16)=(1/2)√2+√2+√2

We know that

⇒2cos^2(π/16)=1+cos(π/8)

⇒2cos^2(π/16)=1+cos√(2+√2/2)

⇒cos^2(π/16)=(2+√2+√3)/4

⇒cos(π/16)=(√2+√2+√3)/2

⇒cos(π/16)=(1/2)√2+√2+√2

Hence cos(π/16)=(1/2)√2+√2+√2 is proved.

Detail Information:-

Prove that cot7(1/2)=√6+√3+√2+2
Prove that cot22(1/2)=√2+1
Prove that cot37(1/2)=√6-√3-√2+2
Prove that tanA/2=√(1-cosA)/(1+cosA)

If sinA=ksinB, Prove that tan(1/2)(A-B)=(k-1)/(k+1)tan(1/2)(A+B)

2021-09-05T12:05:00.000+05:30

Steps to prove if sinA=ksinB, Prove that tan(1/2)(A-B)=(k-1)/(k+1)tan(1/2)(A+B)

From Question given that

⇒sinA=ksinB

⇒sinA/sinB=k/1

⇒(sinA+sinB)/(sinA-sinB)=(K+1)/(k-1)

⇒{2.sin(A+B)/2.cos(A-B)/2}/{2.cos(A+B)/2.cos(A-B)/2}=(K+1)/(k-1)

⇒tan(A+B)/2.cot(A-B)/2=(K+1)/(k-1)

⇒(K-1)/(k+1)tan(A+B)/2=1/cot(A-B)/2

⇒tan(1/2)(A-B)=(k-1)/(k+1)tan(1/2)(A+B)

Hence tan(1/2)(A-B)=(k-1)/(k+1)tan(1/2)(A+B) is proved.

Detail Information:-

If tanA=1/5, tanB=2/3 show that cos2A=cos2B
If A+C=B, Show that the tanA.tanB.tanC=tanB-tanA-tanC
If tanθ=(asinx+bsiny)/(acosx+bcosy) then show that asin(θ–x)+bsin(θ-y)=0
If secA-tanA=1/2 and 0 less than A less than 90 then show that secA=5/4

If acos(x+α)=bcos(x-α) show that (a+b)tanx=(a-b)cotα

2021-09-05T11:56:00.000+05:30

Steps to prove If acos(x+α)=bcos(x-α) show that (a+b)tanx=(a-b)cotα

From Question given that

⇒acos(x+α)=bcos(x-α)

By componendo and dividendo

⇒(a+b)/(a-b)={cos(x-α)+cos(x+α)}/{cos(x+α)+cos(x-α)}

⇒(a+b)/(a-b)=2.cosx.cosα/2.sinx.cosα

⇒(a+b)/(a-b)=cotx.cotα

⇒(1/cotx)(a+b)=(a-b)cotα

⇒(a+b)tanx=(a-b)cotα

Hence (a+b)tanx=(a-b)cotα is proved.

Detail Information:-

If tanA=1/5, tanB=2/3 show that cos2A=cos2B
If A+C=B, Show that the tanA.tanB.tanC=tanB-tanA-tanC
If tanθ=(asinx+bsiny)/(acosx+bcosy) then show that asin(θ–x)+bsin(θ-y)=0
If secA-tanA=1/2 and 0 less than A less than 90 then show that secA=5/4

Prove that cos^3A.cos3A+sin^3A.sin3A=cos^32A

2021-09-05T11:40:00.004+05:30

Steps to prove cos^3A.cos3A+sin^3A.sin3A=cos^32A

L.H.S

=cos^3A.cos3A+sin^3A.sin3A

=cos^3A.(4cos^3A-3cosA)+sin^3A.(3sinA-4sin^3A)

=4cos^6A-3cos^4A+3sin^4A-4sin^6A

=4(cos^6A-sin^6A)-3(cos^4A-sin^4A)

=4{(cos^2A)^3-(sin^2A)^3}-3{(cos^2A)^2-(sin^2A)^2}

=4(cos^2A-sin^2A){(cos^2A)^2+cos^2A.sin^2A+(sin^2A)^2}-3(cos^2A-sin^2A)(cos^2A+sin^2A)

=4(cos^2A-sin^2A)[4{(cos^2A+sin^2A)^2-2cos^2A.sin^2A+cos^2A.sin^2A}-3.1]

=cos2A(4-4sin^2A.cos^2A-3)

=cos2A(1-4sin^2A.cos^2A)

=cos2A(1-sin^2A)

=cos2A.cos^2(2A)=cos^32A

Hence cos^3A.cos3A+sin^3A.sin3A=cos^32A is proved.

Detail Information:-

Express 4cosA.cosB.cosC as the sum of four cosines.
Express cos2A+cos2B+cos2C+cos2(A+B+C) as product of three cosines.
Prove that cos7α+cos3α-cos5α-cosα/sin7α-sin3α-sin5α+sinα=cot2α
Prove that cotA/2=sinA/1-cosA

Prove that cos^6A+sin^6A=(1/4)(1+3cos^2(2A))

2021-09-05T11:25:00.004+05:30

Steps to prove cos^6A+sin^6A=(1/4)(1+3cos^2(2A))

L.H.S

=cos^6A+sin^6A

=(cos^2A)^3+(sin^2A)^3

=(cos^2A-sin^2A){(cos^2A)^2+(sin^2A)^2-cos^2A.sin^2A}

={(cos^2A+sin^2A)^2-2cos^2A.sin^2A-cos^2A.sin^2A}

={1-3sin^2A.cos^2A}

={1-(3/4)(2sinA.cosA)^2}

={1-(3/4)sin^2(2A)}

={1-(3/4)(1-cos^2(2A))}

=1-(3/4)+(3/4)cos^2(2A)

=(4-3)/4+(3/4)cos^2(2A)

=(1/4)(1+3cos^2(2A))

=R.H.S

Hence cos^6A+sin^6A=(1/4)(1+3cos^2(2A)) is proved.

Detail Information:-

Prove that sin^4θ=3/8-(1/2)cos2θ+(1/8)cos4θ
Prove that cot3A=(cot^3A-3cotA)/(3cot^2A-1)
Prove that cot7(1/2)=√6+√3+√2+2
Prove that tan6.tan42.tan66.tan78=1

Prove that sin^4θ=3/8-(1/2)cos2θ+(1/8)cos4θ

2021-09-05T10:09:00.000+05:30

Steps to prove sin^4θ=3/8-(1/2)cos2θ+(1/8)cos4θ

L.H.S

=sin^4θ

=(sin^2θ)^2

={(1-cos2θ)/2}^2

=(1/4)(1-cos2θ)^2

=(1/4)[(1)^2+cos^2(2θ)-2.1.cos2θ]

=(1/4){1-2cos2θ+(1+cos4θ)/2}

=1/4-2cos2θ/4+(1+cos4θ)/8

=1/4-2cos2θ/4+1/8+cos4θ/8

=(2+1-4cos2θ+cos4θ)/8

=3/8-(1/2)cos2θ+(1/8)cos4θ

=R.H.S

Hence sin^4θ=3/8-(1/2)cos2θ+(1/8)cos4θ is proved.

Detail Information:-

Prove that sin2A+sin2B/sin2A-sin2B=tan(A+B)/tan(A-B)
Prove that sin2A+sin5A-sinA/cos2A+cos5A+cosA=tan2A
Prove that 8sin10.sin50.sin70=1
Prove that 4sinA.sin(60-A).sin(60+A)

Prove that cot3A=(cot^3A-3cotA)/(3cot^2A-1)

2021-09-05T09:59:00.001+05:30

Steps to prove cot3A=(cot^3A-3cotA)/(3cot^2A-1)

L.H.S

=cot3A

=cot(2A+A)

=(cot2A.cotA-1)/(cotA+cot2A)

={((cot^2A-1)/2cotA).cotA-1}/{cotA+(cot^2A-1)/2cotA)}

={(cot^3A-cotA-2cotA)/2cotA}/{(2cot^2A+cot^2A-1)/cot2A}

=(3cot^3A-3cotA)/(3cot^2A-1)

=R.H.S

Hence cot3A=(cot^3A-3cotA)/(3cot^2A-1) is proved.

Detail Information:-

Prove that cotA/2=sinA/1-cosA
Prove cotA-cosec2A=cot2A
Prove that cotA-tanA=2cot2A
Prove that cotA-tanA/cotA+tanA=cos2A

Prove that cot7(1/2)=√6+√3+√2+2

2021-09-05T09:50:00.004+05:30

Steps to prove cot7(1/2)=√6+√3+√2+2

L.H.S

=cot7(1/2)

=cot(15/2)

=(1+sin15)/cos15

={1+(√3+1/2√2)}/(√3-1/2√2)

=(2√2+√3+1)/(√3-1)

=(2√2+√3+1)(√3+1)/(√3-1)(√3+1)

=(2√6+2√3+2√2+4)/3-1

=(2√6+2√3+2√2+4)/2

=2(√6+√3+√2+2)/2

=√6+√3+√2+2

=R.H.S

Hence cot7(1/2)=√6+√3+√2+2 is proved.

Detail Information:-

Prove that cot22(1/2)=√2+1

2021-09-05T09:43:00.004+05:30

Steps to prove cot22(1/2)=√2+1

L.H.S

=cot22(1/2)

=cot45/2

=(1+cos45)/sin45

={1+(1/√2)}/(1/√2)

={(√2+1)/2}/1/√2

=√2+1

=R.H.S

Hence cot22(1/2)=√2+1 is proved.

Detail Information:-

Prove that cot22(1/2)=√2+1
Prove that cotA-tanA/cotA+tanA=cos2A
Prove that cotA-tanA=2cot2A
Prove cotA-cosec2A=cot2A

Prove that cot37(1/2)=√6-√3-√2+2

2021-09-05T09:36:00.002+05:30

Steps to prove cot37(1/2)=√6-√3-√2+2

L.H.S

=cot37(1/2)

=cot(75/2)

=(1+cos75)/sin75

={1+(√3-1/2√2)}/(√3+1/2√2)

=(2√2+√3-1)/(√3+1)

=(2√2+√3-1)(√3-1)/(√3+1)(√3-1)

=(2√6-2√3-2√2+4)/3-1

=(2√6-2√3-2√2+4)/2

=2(√6-√3-√2+2)/2

=√6-√3-√2+2

=R.H.S

Hence cot37(1/2)=√6-√3-√2+2 is proved.

Detail Information:-

Prove that cotA-tanA=2cot2A
Prove that cotA-tanA/cotA+tanA=cos2A
Prove that cot2A=cot^2A-1/2cotA
Prove that tan75+cot75=4

Prove that tan6.tan42.tan66.tan78=1

2021-09-05T09:25:00.000+05:30

Steps to prove tan6.tan42.tan66.tan78=1

L.H.S

=tan6.tan42.tan66.tan78

=tan6.tan(60-6).tan(60+6).tan42.tan78/tan54

=tan3(6).tan42.tan78/tan54

=tan18.tan42.tan78/tan54

=tan18.tan(60-18).tan(60+18)/tan54

=tan3(18)/tan54

=tan54/tan54

=1

=R.H.S

Hence tan6.tan42.tan66.tan78=1 is proved.

Detail Information:-

Prove that tanθ(1+sec2θ)=tan2θ
Prove that tanA/2=√(1-cosA)/(1+cosA)
Prove that tan3A-tan2A-tanA=tan3A.tan2A.tanA
Prove that cotA-tanA/cotA+tanA=cos2A

Read-Fx

Prove that cos^2(A/2)+cos^2(B/2)+cos^2(C/2)=2.cos(A/2)cos(B/2).sin(C/2)

Steps to prove cos^2(A/2)+cos^2(B/2)+cos^2(C/2)=2.cos(A/2)cos(B/2).sin(C/2)

L.H.S

=cos^2(A/2)+cos^2(B/2)+cos^2(C/2)

=(1+cosA)/2+(1+cosB)/2-(1+cosC)/2

=(1+cosA+1+cosB-1-cosC)/2

=(1+cosA+cosB-cosC)/2

=1/2+(cosA+cosB-cosC)/2

=1/2+1/2{2.sin(C/2).cos(A-B)/2-1+2sin^2(C/2)}

=1/2+1/2[2.sin(C/2).{cos(A-B)/2+sin(C/2)}-1]

=1/2+1/2[2.sin(C/2).2cos(A/2).cos(B/2)-1]

=1/2+2.cos(A/2)cos(B/2).sin(C/2)-1/2

=2.cos(A/2)cos(B/2).sin(C/2)

R.H.S

Hence cos^2(A/2)+cos^2(B/2)+cos^2(C/2)=2.cos(A/2)cos(B/2).sin(C/2) is proved.

Detail Information:-

Similar Questions For You:-

Prove that cos^3A.cos3A+sin^3A.sin3A=cos^32AProve that cos^6A+sin^6A=(1/4)(1+3cos^2(2A))Prove that sin^4θ=3/8-(1/2)cos2θ+(1/8)cos4θProve that cot3A=(cot^3A-3cotA)/(3cot^2A-1)

If x+y+z=xyz then prove that x/(1-x^2)+y/(1-y^2)+z/(1-z^2)=xyz/{(1-x^2)(1-y^2)(1-z^2)}

Steps to prove x/(1-x^2)+y/(1-y^2)+z/(1-z^2)=xyz/{(1-x^2)(1-y^2)(1-z^2)}

Let us consider

x=tanα, y=tanβ, z=tanγ

⇒x+y+z=xyz

⇒tanα+tanβ+tanγ=tanα.tanβ.tanγ

⇒tanα+tanβ=tanα.tanβ.tanγ-tanγ

⇒tanα+tanβ=tanγ(tanα.tanβ-1)

⇒(tanα+tanβ)/(1-tanα.tanβ)=-tanγ

⇒tan(α+β)=-tanγ

⇒tan(α+β)=tan(pi-γ)

⇒α+β=pi-γ

⇒2α+2β=2pi-2γ

⇒tan(2α+2β)=tan(2pi-2γ)

⇒(tan2α+tan2β)/(1-tan2α.tan2β)=-tan2γ

⇒tan2α+tan2β+tan2γ=tan2α.tan2β.tan2γ

⇒2x/(1-x^2)+2y/(1-y^2)+2z/(1-z^2)=2x/(1-x^2).2y/(1-y^2).2z/(1-z^2)

⇒x/(1-x^2)+y/(1-y^2)+z/(1-z^2)=xyz/{(1-x^2)(1-y^2)(1-z^2)}

Hence x/(1-x^2)+y/(1-y^2)+z/(1-z^2)=xyz/{(1-x^2)(1-y^2)(1-z^2)} is proved.

Detail Information:-

Similar Questions For You:-

Prove that sin^2(A/2)+sin^2(B/2)+sin^2(C/2)=1-2.sin(A/2)sin(B/2).sin(C/2)

Steps to prove sin^2(A/2)+sin^2(B/2)+sin^2(C/2)=1-2.sin(A/2)sin(B/2).sin(C/2)

L.H.S

=sin^2(A/2)+sin^2(B/2)+sin^2(C/2)

=(1-cosA)/2+(1-cosB)/2+(1-cosC)/2

=(1-cosA+1-cosB+1-cosC)/2

={3-(cosA+cosB+cosC)}/2

=(1/2)[3-{2.cos(A+B)/2.cos(A-B)/2+1-2sin^2(c/2)}]

=(1/2)[3-2.cos(A+B)/2.cos(A-B)/2-1+2sin^2(c/2)}]

=(1/2)[2-2sin(C/2){cos(A-B)/2-cos(A+B)/2}]

=(1/2).2[1-sin(C/2){cos(A-B)/2-cos(A+B)/2}]

=1-sin(C/2){cos(A/2-B/2)-cos(A/2+B/2)}

=1-2.sin(A/2)sin(B/2).sin(C/2)

Hence sin^2(A/2)+sin^2(B/2)+sin^2(C/2)=1-2.sin(A/2)sin(B/2).sin(C/2) is proved.

Detail Information:-

Similar Questions For You:-

An angle θ is divide into two parts α,β such that tanα:tanβ=x:y then prove that sin(α-β)={(x-y)/(x+y)}sinθ

Steps to prove sin(α-β)={(x-y)/(x+y)}sinθ

From Question We know that

⇒tanα:tanβ=x:y

⇒tanα/tanβ=x/y

⇒(sinα/cosα)/(sinβ/cosβ)=x/y

⇒(sinα.cosβ)/(cosβ.sinα)=x/y

By componendo and dividendo

⇒{(sinα.cosβ)+(cosα.sinβ)}/{(cosβ.sinα)-(cosα.sinβ)}=(x+y)/(x-y)

⇒sin(α+β)/sin(α-β)=(x+y)/(x-y)

⇒sinθ/sin(α-β)=(x+y)/(x-y)

⇒sin(α-β)={(x-y)/(x+y)}sinθ

Hence sin(α-β)={(x-y)/(x+y)}sinθ is proved.

Detail Information:-

Similar Questions For You:-

If (1-e)tan^2(β/2)=(1+e)tan^2(α/2), then prove that cosβ=(cosα-e)/(1-ecosα)If tanA+tanB=a and cotA+cotB=b, then prove that cot(A+B)=1/a-1/bIf sec(φ+α)+sec(φ-α)=2secφ show that cosφ=+-√cosα/2If A+B+C= π than prove that sin2A+sin2B-sin2C=4.cosA.cosB.sinC

If (1-e)tan^2(β/2)=(1+e)tan^2(α/2), then prove that cosβ=(cosα-e)/(1-ecosα)

Steps to prove cosβ=(cosα-e)/(1-ecosα)

From Question

⇒(1-e)tan^2(β/2)=(1+e)tan^2(α/2)

⇒tan^2(β/2)={(1+e)/(1-e)}tan^2(α/2)

L.H.S

=cosβ

={1-tan^2(β/2)}/{1+tan^2(β/2)}

Prove that cos^3A.cos3A+sin^3A.sin3A=cos^32A
Prove that cos^6A+sin^6A=(1/4)(1+3cos^2(2A))
Prove that sin^4θ=3/8-(1/2)cos2θ+(1/8)cos4θ
Prove that cot3A=(cot^3A-3cotA)/(3cot^2A-1)

If (1-e)tan^2(β/2)=(1+e)tan^2(α/2), then prove that cosβ=(cosα-e)/(1-ecosα)
If tanA+tanB=a and cotA+cotB=b, then prove that cot(A+B)=1/a-1/b
If sec(φ+α)+sec(φ-α)=2secφ show that cosφ=+-√cosα/2
If A+B+C= π than prove that sin2A+sin2B-sin2C=4.cosA.cosB.sinC

If tanA+tanB=a and cotA+cotB=b, then prove that cot(A+B)=1/a-1/b
If sec(φ+α)+sec(φ-α)=2secφ show that cosφ=+-√cosα/2
If A+B+C= π than prove that sin2A+sin2B-sin2C=4.cosA.cosB.sinC
If A+B+C= π than prove that cosA+cosB+cosC=1+4.sin(1/2)A.sin(1/2)B.sin(1/2)C

If sec(φ+α)+sec(φ-α)=2secφ show that cosφ=+-√cosα/2
If A+B+C= π than prove that sin2A+sin2B-sin2C=4.cosA.cosB.sinC
If A+B+C= π than prove that sinA+sinB-sinC=4.sin(1/2)A.sin(1/2)B.cos(1/2)C
If A+B+C= π than prove that cosA+cosB+cosC=1+4.sin(1/2)A.sin(1/2)B.sin(1/2)C

If α,β are acute angles and cos2α=(3cos2β-1)/(3-cos2β) than prove that tanα=√2tanβ
If 2tanα=3tanβ than prove that tanβ/(2+3tan^2β)
If acos(x+α)=bcos(x-α) show that (a+b)tanx=(a-b)cotα
Prove that cos7α+cos3α-cos5α-cosα/sin7α-sin3α-sin5α+sinα=cot2α