Question

use a double - angle identity to find the exact value of the expression.
\\( \cos ^ { 2 } 157.5 ^ { \circ } - \sin ^ { 2 } 157.5 ^ { \circ } \\)
\\( \cos ^ { 2 } 157.5 ^ { \circ } - \sin ^ { 2 } 157.5 ^ { \circ } = \square \\)
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expressio

Explanation:

Step1: Apply double - angle identity

The double - angle identity for cosine is $\cos2\alpha=\cos^{2}\alpha - \sin^{2}\alpha$.
Let $\alpha = 157.5^{\circ}$, then $\cos^{2}157.5^{\circ}-\sin^{2}157.5^{\circ}=\cos(2\times157.5^{\circ})$.

Step2: Calculate the angle

Calculate $2\times157.5^{\circ}=315^{\circ}$.
So, $\cos(2\times157.5^{\circ})=\cos315^{\circ}$.

Step3: Find the value of cosine

We know that $\cos315^{\circ}=\cos(360^{\circ}-45^{\circ})$.
Using the identity $\cos(A - B)=\cos A\cos B+\sin A\sin B$ (here $A = 360^{\circ},\cos360^{\circ}=1,\sin360^{\circ}=0,B = 45^{\circ},\cos45^{\circ}=\frac{\sqrt{2}}{2},\sin45^{\circ}=\frac{\sqrt{2}}{2}$), or using the unit - circle definition. Since $\cos\theta=x$ on the unit circle $(x,y)=\cos\theta + i\sin\theta$, and for $\theta = 315^{\circ}$, the reference angle is $45^{\circ}$ and it is in the fourth quadrant where cosine is positive. So, $\cos315^{\circ}=\frac{\sqrt{2}}{2}$.

Answer:

$\frac{\sqrt{2}}{2}$