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 $\cos x \ne 0$:

\(\ds \cos^2 x + \sin^2 x\) \(=\) \(\ds 1\) Sum of Squares of Sine and Cosine
\(\ds \leadsto \ \ \) \(\ds 1 + \frac {\sin^2 x} {\cos^2 x}\) \(=\) \(\ds \frac 1 {\cos^2 x}\) 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

$\blacksquare$


Also presented as

Difference of Squares of Secant and Tangent can also be presented 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

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