# Sum of Squares of Sine and Cosine/Corollary 1

< Sum of Squares of Sine and Cosine(Redirected from Difference of Squares of Secant and Tangent)

## 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

- 1964: Murray R. Spiegel:
*Theory and Problems of Complex Variables*... (previous) ... (next): $2$: The Elementary Functions: $4$ - 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.20$