Sum of Squares of Sine and Cosine/Corollary 1

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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 $\quad$
\(\displaystyle \implies \ \ \) \(\displaystyle 1 + \frac {\sin^2 x} {\cos^2 x}\) \(=\) \(\displaystyle \frac 1 {\cos^2 x}\) $\quad$ dividing both sides by $\cos^2 x$, as $\cos x \ne 0$ $\quad$
\(\displaystyle \implies \ \ \) \(\displaystyle 1 + \tan^2 x\) \(=\) \(\displaystyle \sec^2 x\) $\quad$ Definitions of tangent and secant $\quad$
\(\displaystyle \implies \ \ \) \(\displaystyle \sec^2 x - \tan^2 x\) \(=\) \(\displaystyle 1\) $\quad$ 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 $\cos x \ne 0$)}$


Sources