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