1 sin 2 α−cos 2 α−1

To simplify the expression \( \sin^2 \alpha - \cos^2 \alpha - 1 \), we can use the Pythagorean identity, which states that:

\[ \sin^2 \alpha + \cos^2 \alpha = 1 \]

Rearranging this, we find:

\[ \sin^2 \alpha = 1 - \cos^2 \alpha \]

Now, substitute \( \sin^2 \alpha \) back into the original expression:

\[ \sin^2 \alpha - \cos^2 \alpha - 1 \] \[ = (1 - \cos^2 \alpha) - \cos^2 \alpha - 1 \]

Now simplify:

\[ = 1 - \cos^2 \alpha - \cos^2 \alpha - 1 \] \[ = -2\cos^2 \alpha \]

Thus, the simplified expression is:

\[ \sin^2 \alpha - \cos^2 \alpha - 1 = -2\cos^2 \alpha \]