How can I prove this trigonometric statement true?

Question

$$ {1+\sin^{2}\left(x\right) \over \cos^{2}\left(x\right)} = 1 + 2\tan^{2}\left(x\right)$$

This statement is part of a larger problem, but I need to prove that this is true before moving on. I'm assuming that I would first need to prove $1 + \sin^{2}\left(x\right) = 2\sin^{2}\left(x\right) + \cos^{2}\left(x\right)$, but I'm not sure how to prove that. Any help would be greatly appreciated !.

Answer 1

$$\frac{1+\sin^2x}{\cos^2x}=\frac1{\cos^2x}+\frac{\sin^2x}{\cos^2x}$$

$$=\sec^2x+\tan^2x=(1+\tan^2x)+\tan^2x$$

or

$$\frac{1+\sin^2x}{\cos^2x}=\frac{\cos^2x+\sin^2x+\sin^2x}{\cos^2x}=\cdots$$

Answer 2

Multiply through by $\cos^2(x)$ to get $$ 1+\sin^2(x)=\cos^2(x)+2\sin^2(x) $$ then use $\sin^2(x)+\cos^2(x)=1$.