# Sum of Squares of Sine and Cosine/Corollary 2

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## Corollary to Sum of Squares of Sine and Cosine

For all $x \in \C$:

- $\csc^2 x - \cot^2 x = 1 \quad \text {(when $\sin x \ne 0$)}$

where $\csc$, $\cot$ and $\sin$ are cosecant, cotangent and sine respectively.

## Proof

When $\sin x \ne 0$:

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

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

\(\displaystyle \leadsto \ \ \) | \(\displaystyle 1 + \cot^2 x\) | \(=\) | \(\displaystyle \csc^2 x\) | Definition of Cotangent and Definition of Cosecant | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \csc^2 x - \cot^2 x\) | \(=\) | \(\displaystyle 1\) | rearranging |

$\blacksquare$

## Also defined as

This result can also be reported as:

- $\csc^2 x = 1 + \cot^2 x \quad \text{(when $\sin x \ne 0$)}$

or:

- $\cot^2 x = \csc^2 x - 1 \quad \text{(when $\sin x \ne 0$)}$

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.21$ - 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: The Elementary Functions: $4$