Are any of these three problems identities?

Question

1. Cos^2x-sin^2x=1-2sin^2x
2. Sinxsecx=cosxcscx
3. Sec^4x-tan^4/sec^2x=1+sin^2x
If so, how can you conclude that any of them are identities?

Answer 1

Yes. You can prove an identity by rearranging one side of the equation to match the other.

For example:
1.
cos^2(x) - sin^2(x) = 1 - sin^2(x) - sin^2(x) because cos^2(x) + sin^2(x) = 1
= 1 - 2 sin^2(x)

Answer 2

The second one is not an identity. You can prove this by substituting the appropriate sin/cos functions for sec(x) and csc(x).