Question

use an identity to find the value of the following expression. do not use a calculator.

\\( \sin ^ { 2 } \frac { \pi } { 12 } + \cos ^ { 2 } \frac { \pi } { 12 } \\)

\\( \sin ^ { 2 } \frac { \pi } { 12 } + \cos ^ { 2 } \frac { \pi } { 12 } = \square \\) (type a whole number.)

Explanation:

Step1: Recall Pythagorean identity

The Pythagorean identity in trigonometry states that for any angle \(\theta\), \(\sin^{2}\theta+\cos^{2}\theta = 1\).

Step2: Apply the identity to the given expression

In the given expression \(\sin^{2}\frac{\pi}{12}+\cos^{2}\frac{\pi}{12}\), we can let \(\theta=\frac{\pi}{12}\). Then by the Pythagorean identity \(\sin^{2}\theta+\cos^{2}\theta=1\), substituting \(\theta = \frac{\pi}{12}\) we get \(\sin^{2}\frac{\pi}{12}+\cos^{2}\frac{\pi}{12}=1\).

Answer:

\(1\)