Question

How do you simplify the expression by using a double-angle formula $2{\cos }^{2}5\theta -1$?

Answer 1

Simplify the expression:

From the double-angle formula of $\cos \theta$:

$\mathbf{cos}\mathbf{2}\mathbf{x}\mathbf{=}\mathbf{2}{\mathbf{cos}}^{\mathbf{2}}\mathbf{x}\mathbf{-}\mathbf{1}$

Let, $x=5\theta$.

Now, the formula can be rewritten as:

$$\begin{gathered} \cos 2\left(5\theta \right)=2{\cos }^{2}5\theta -1 \\ \Rightarrow \cos 10\theta =2{\cos }^{2}5\theta -1 \end{gathered}$$

Hence, the simplified expression of $2{\cos }^{2}5\theta -1$ is $\cos 10\theta$.