Chapter 7, Problem 67RE

To determine

To establish:the given identity.

Answer to Problem 67RE

The identity is established.

Explanation of Solution

Given:

The given identity is $1-8{\sin }^2\theta {\cos }^2\theta =\cos \left(4\theta \right)$ .

Calculation:

As the given identity is $1-8{\sin }^2\theta {\cos }^2\theta =\cos \left(4\theta \right)$ .

Now, let us check RHS:

  $\begin{array}{l}RHS=\cos \left(4\theta \right)\\ \left\{\text{since,}\left(1-2{\sin }^2\theta \right)=\cos 2\theta \right\}\\ =1-2{\sin }^2\left(2\theta \right)\\ =1-2{\left(\sin \left(2\theta \right)\right)}^2\\ =1-2{\left(2\sin \theta cos\theta \right)}^2\left\{\sin 2A=2\sin A\cos A\right\}\\ =1-2\left(4{\sin }^2\theta co{s}^2\theta \right)\\ =1-8{\sin }^2\theta co{s}^2\theta \\ =LHS\end{array}$

Thus, $1-8{\sin }^2\theta {\cos }^2\theta =\cos \left(4\theta \right)$

Hence, the identity is established.