Question

How many values of $\theta ϵ[0,\frac{\pi }{2}]$, satisfy the relation cos $\theta +cos3\theta +cos5\theta +cos7\theta =0$ ?

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Answer 1

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.

$\left[\right(cos\theta +cos3\theta )+(cos5\theta +cos7\theta )$ is also fine]
$\Rightarrow 2cos4\theta cos3\theta +2cos\theta cos4\theta =\theta$

$\Rightarrow cos4\theta (cos\theta +cos3\theta )=0$

$\Rightarrow cos4\theta 2cos\theta cos2\theta =0$

$\Rightarrow cos\theta cos2\theta cos4\theta =0$

$\Rightarrow cos\theta =0orcos2\theta =0orcos4\theta =0$

We want only those values of $\theta ϵ[0,\frac{\pi }{2}]$
$\Rightarrow \theta =\frac{\pi }{2}or2\theta =\frac{\pi }{2}or4\theta =\frac{\pi }{2}or\frac{3\pi }{2}$
$\Rightarrow \theta =\frac{\pi }{2}or\theta =\frac{\pi }{4}or\theta =\frac{\pi }{8}or\theta =\frac{3\pi }{8}$
$\Rightarrow$ 4 values