Sum of Squares of Sine and Cosine/Corollary 1

Corollary to Sum of Squares of Sine and Cosine

For all $x \in \C$:

$\sec^2 x - \tan^2 x = 1 \quad \text {(when $\cos x \ne 0$)}$

where $\sec$, $\tan$ and $\cos$ are secant, tangent and cosine respectively.

When $\cos x \ne 0$:

$\ds \cos^2 x + \sin^2 x$

$=$

$\ds 1$

$\ds \leadsto \ \ $

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

$=$

$\ds \frac 1 {\cos^2 x}$

dividing both sides by $\cos^2 x$, as $\cos x \ne 0$

$\ds \leadsto \ \ $

$\ds 1 + \tan^2 x$

$=$

$\ds \sec^2 x$

Definition of Tangent Function and Definition of Secant Function

$\ds \leadsto \ \ $

$\ds \sec^2 x - \tan^2 x$

$=$

$\ds 1$

rearranging

$\blacksquare$

This result can also be reported as:

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

or:

$\tan^2 x = \sec^2 x - 1 \quad \text {(when $\cos x \ne 0$)}$