If α equals to π by 2^n + 1, prove that 2^nCosα cos 2α cos 2^2 α .......... cos 2^n-1 α = 1

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Question no. 4

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Answer 1

Solution: you can just replace x by alpha according to your question

Explanation : Observe that there are n terms of cos, and, in 2ⁿ,

the no. of 2 is

also n. So, we utilise each 2 with every term of cos.

We will also use the identity :sin2θ=2sinθcosθ. Thus,

The L.H.S.=(2cosx)(2cos2x)(2cos4x)(2cos8x)...(2cos2ⁿ⁻¹x)

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But, x=π/2ⁿ+1⇒2ⁿx+x=π,or,2ⁿx=π−x, so that,

sin(2ⁿx)=sin(π−x)=sinx. Therefore,

The L.H.S.=sinx/sinx=1=The R.H.S

Hence proved.