Question

show that (Cos 7theta- iSin 7theta)² (Cos 3theta+ iSin 3theta)⁴ /(cos 3theta+ iSin 5theta)³ (Cos theta- iSin theta)^7 =1​

Answer 1

Answer:

You have limited access to toppr answr. Sign to unlock 35 lakh questions

What's your doubt?

Maths

Bookmark

Solve:

cosθ+cos7θ+cos3θ+cos5θ=0

share

Share

Answer

We have,

cosθ=cos7θ=cos3θ+cos5θ=0

We know that,

cosC+cosD=2cos

2

C+D

cos

2

C−D

Now,

cosθ+cos7θ+cos3θ+cos5θ=0

2cos

2

θ+7θ

cos

2

θ−7θ

+2cos

2

3θ+5θ

cos

2

3θ−5θ

=0

2cos4θcos(−3θ)+2cos4θcos(−θ)=0

2cos4θ[cos3θ+cosθ]=0

∴cos(−θ)=cosθ

Again,

2cos4θ[2cos

2

3θ+θ

cos

2

3θ−θ

]=0

2cos4θ[2cos2θcosθ]=0

4cos4θcos2θcosθ=0

cos4θcos2θcosθ=0

If cos4θ=0

cos4θ=cos90

o

cos4θ=cos

2

π

4θ=

2

π

θ=

8

π

If cos2θ=0

cos2θ=cos

2

π

2θ=

2

π

θ=

4

π

If cosθ=0

cosθ=cos

2

π

θ=

2

π

Hence, θ=

2

π

,

4

π

,

8

π