For which angles $a$ is $\sin^4 a - \cos^4 a > \sin^2 a - \cos^2 a$?

Question

My questions

1. For which angles $a$ is $\sin^4 a - \cos^4 a > \sin^2 a - \cos^2 a$?

2. For which angles $a$ is $\sin^4 a - \cos^4 a \ge \sin^2 a - \cos^2 a$?

I understand that the two sides will be equal at $90$ degrees and at $45$ degrees. And I understand that between $45$ and $90$ degrees the first inequality is valid. Below $45$ degrees it isn't. Is this correct? Thanks.

Answer 1

Hints:

$$\sin^2\alpha+\cos^2\alpha=1.$$

Answer 2

For the first case, there is no such $\alpha$ that suffices the inequality.

For the second case, it suffices for all $\alpha$.

This is proven when you use $sin^2 + cos^2 = 1$ to change both sides and see that both sides are actually saying the same thing.