Question

use an identity to simplify the expression. do not use a calculator.

\\( \cos ^ { 2 } \left( \frac { \pi } { 11 } \
ight) - \sin ^ { 2 } \left( \frac { \pi } { 11 } \
ight) \\)

\\( \cos ^ { 2 } \left( \frac { \pi } { 11 } \
ight) - \sin ^ { 2 } \left( \frac { \pi } { 11 } \
ight) = \square \\)
(type an exact answer, using \\( \pi \\) as needed.)

Explanation:

Step1: Recall the double - angle identity

The double - angle identity for cosine is $\cos(2\alpha)=\cos^{2}\alpha-\sin^{2}\alpha$.
Let $\alpha = \frac{\pi}{11}$. Then the given expression $\cos^{2}(\frac{\pi}{11})-\sin^{2}(\frac{\pi}{11})$ can be rewritten using the identity.

Step2: Substitute the value of $\alpha$

Substituting $\alpha=\frac{\pi}{11}$ into the identity $\cos(2\alpha)=\cos^{2}\alpha - \sin^{2}\alpha$, we get $\cos(2\times\frac{\pi}{11})$.
So, $\cos^{2}(\frac{\pi}{11})-\sin^{2}(\frac{\pi}{11})=\cos(\frac{2\pi}{11})$.

Answer:

$\cos(\frac{2\pi}{11})$