Question

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

Answer 1

Recall the identity

`\cos^2\Big((\beta)/(2)\Big)=(1+\cos\beta)/(2)`

We can rewrite the given equation into

`(\cos(5x))^2\Longrightarrow\cos^2\Big(5x\Big)\Longrightarrow\cos^2\Big((10x)/(2)\Big)`

Using the identity we get

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

Therefore,

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