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.

When $\sin x \ne 0$:

$\ds \sin^2 x + \cos^2 x$

$=$

$\ds 1$

$\ds \leadsto \ \ $

$\ds 1 + \frac {\cos^2 x} {\sin^2 x}$

$=$

$\ds \frac 1 {\sin^2 x}$

dividing both sides by $\sin^2 x$, as $\sin x \ne 0$

$\ds \leadsto \ \ $

$\ds 1 + \cot^2 x$

$=$

$\ds \csc^2 x$

Definition of Cotangent and Definition of Cosecant

$\ds \leadsto \ \ $

$\ds \csc^2 x - \cot^2 x$

$=$

$\ds 1$

rearranging

$\blacksquare$

$\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 $\sin x \ne 0$)}$

1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: Formulae $(3)$
1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: Definitions of the ratios
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: $4$
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): trigonometric function
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): trigonometric function
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $12$: Trigonometric formulae
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $14$: Trigonometric formulae