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.

Proof

When $\cos x \ne 0$:

 $\displaystyle \cos^2 x + \sin^2 x$ $=$ $\displaystyle 1$ $\quad$ Sum of Squares of Sine and Cosine $\quad$ $\displaystyle \leadsto \ \$ $\displaystyle 1 + \frac {\sin^2 x} {\cos^2 x}$ $=$ $\displaystyle \frac 1 {\cos^2 x}$ $\quad$ dividing both sides by $\cos^2 x$, as $\cos x \ne 0$ $\quad$ $\displaystyle \leadsto \ \$ $\displaystyle 1 + \tan^2 x$ $=$ $\displaystyle \sec^2 x$ $\quad$ Definition of Tangent and Definition of Secant $\quad$ $\displaystyle \leadsto \ \$ $\displaystyle \sec^2 x - \tan^2 x$ $=$ $\displaystyle 1$ $\quad$ rearranging $\quad$

$\blacksquare$

Also defined as

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$)}$