# 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 $\cos x \ne 0$:

\(\displaystyle \cos^2 x + \sin^2 x\) | \(=\) | \(\displaystyle 1\) | Sum of Squares of Sine and Cosine | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle 1 + \frac {\sin^2 x} {\cos^2 x}\) | \(=\) | \(\displaystyle \frac 1 {\cos^2 x}\) | dividing both sides by $\cos^2 x$, as $\cos x \ne 0$ | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle 1 + \tan^2 x\) | \(=\) | \(\displaystyle \sec^2 x\) | Definition of Tangent Function and Definition of Secant Function | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \sec^2 x - \tan^2 x\) | \(=\) | \(\displaystyle 1\) | rearranging |

$\blacksquare$

## 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

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.20$ - 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: The Elementary Functions: $4$