(g.) prove that sin2x - cos2x = -1

From this identity, we can express $\sin^2 x$ as $1 - \cos^2 x$.

The statement requires us to prove $1 - 2\cos^2 x = -1$.

Add $2\cos^2 x$ to both sides and add 1 to both sides:

Divide by 2:

Take the square root of both sides:

This means that the original statement $\sin^2 x - \cos^2 x = -1$ is only true when $\cos x = 1$ (i.e., $x = 2n\pi$ for any integer $n$) or $\cos x = -1$ (i.e., $x = (2n+1)\pi$ for any integer $n$). In other words, it is true when $x$ is an integer multiple of $\pi$.

Conclusion: The statement $\sin^2 x - \cos^2 x = -1$ is not a trigonometric identity that holds true for all values of $x$. It is only true for specific values of $x$ where $x = n\pi$ for any integer $n$. Therefore, it cannot be proven as a general identity.