Sum of Squares of Sine and Cosine/Corollary 1

Theorem

 * $(1): \quad 1 + \tan^2 x = \sec^2 x \quad \text{(when $\cos x \ne 0$)}$
 * $(2): \quad 1 + \cot^2 x = \csc^2 x \quad \text{(when $\sin x \ne 0$)}$

where:


 * $\tan$ and $\cot$ are tangent and cotangent.
 * $\sec$ and $\csc$ are secant and cosecant.

Proof
Proof of $(1)$, when $\cos x \ne 0$:

Proof of $(2)$, when $\sin x \ne 0$: