Question

1. $\cos ^{2}y-\sin ^{2}y=1-2\sin ^{2}y$

Explanation:

Step1: Use the Pythagorean identity

We know that \(\cos^{2}y+\sin^{2}y = 1\), so \(\cos^{2}y=1 - \sin^{2}y\).

Step2: Substitute \(\cos^{2}y\) in the left - hand side

Substitute \(\cos^{2}y = 1-\sin^{2}y\) into \(\cos^{2}y-\sin^{2}y\). Then we get \((1 - \sin^{2}y)-\sin^{2}y\).

Step3: Simplify the expression

\((1 - \sin^{2}y)-\sin^{2}y=1-\sin^{2}y-\sin^{2}y=1 - 2\sin^{2}y\)

Answer:

The left - hand side \(\cos^{2}y-\sin^{2}y\) is equal to the right - hand side \(1 - 2\sin^{2}y\) after using the Pythagorean identity \(\cos^{2}y+\sin^{2}y = 1\) and simplifying. So the identity \(\cos^{2}y-\sin^{2}y=1 - 2\sin^{2}y\) is verified.