Answer
$$\sin^4\theta-\cos^4\theta=2\sin^2\theta-1$$
The equation has been verified to be an identity.
Work Step by Step
$$\sin^4\theta-\cos^4\theta=2\sin^2\theta-1$$
We would simplify the left side first, seeing that it has up to fourth factors: $$\sin^4\theta-\cos^4\theta$$ $$=(\sin^2\theta-\cos^2\theta)(\sin^2\theta+\cos^2\theta)$$ $\sin^2\theta+\cos^2\theta=1$ is an identity, which means $$=(\sin^2\theta-\cos^2\theta)\times1$$ $$=\sin^2\theta-\cos^2\theta$$
Now, a trick here in the identity proving exercise is to see whether one side has but the other does not have (yet) and tries to add the same thing and sees how it will turn out.
For example, in this problem, until now we see that the right side has $2\sin^2\theta$, but the left side only has $\sin^2\theta$, so we would try adding $1 \sin^2\theta$ to the left side, meaning that $$\sin^2\theta-\cos^2\theta$$ $$=(\sin^2\theta+\sin^2\theta)-\sin^2\theta-\cos^2\theta$$ (remember that adding also comes along with subtracting so that the value of the formula remains the same) $$=2\sin^2\theta-(\sin^2\theta+\cos^2\theta)$$ $$=2\sin^2\theta-1$$ (again, $\sin^2\theta+\cos^2\theta=1$)
The identity is thus verified.
