Sum of Squares of Sine and Cosine/Corollary 2

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$)}$