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Question

How many values of θϵ[0,π2], satisfy the relation cos θ+cos3θ+cos5θ+cos7θ=0 ?


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Solution

When we see such expressions, we try to use the transformation formula to simplify them. We will

combine the terms such that we will get common terms.

[(cosθ+cos3θ)+(cos5θ+cos7θ) is also fine]
2cos4θcos3θ+2cosθcos4θ=θ

cos4θ(cosθ+cos3θ)=0

cos4θ2cosθcos2θ=0

cosθcos2θcos4θ=0

cosθ=0orcos2θ=0 or cos4θ=0

We want only those values of θϵ[0,π2]
θ=π2or2θ=π2or4θ=π2or3π2
θ=π2orθ=π4orθ=π8orθ=3π8
4 values


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