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