Question

Use a half angle formula to fill in the blanks in the identity below

Answer 1

Recall the identity

$\cos^2\Big(\frac{\beta}{2}\Big)=\frac{1+\cos\beta}{2}$

We can rewrite the given equation into

$(\cos(5x))^2\Longrightarrow\cos^2\Big(5x\Big)\Longrightarrow\cos^2\Big(\frac{10x}{2}\Big)$

Using the identity we get

$\begin{gathered} \beta=10x \\ \\ \cos^2\Big(\frac{10x}{2}\Big)=\frac{1+\cos(10x)}{2} \\ \cos^2\Big(\frac{10x}{2}\Big)=\frac{1}{2}+\frac{1}{2}\cos(10x) \end{gathered}$

Therefore,

$(\cos(5x))^2=0.5+0.5\cos(10x)$