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