Sum of Squares of Sine and Cosine/Corollary 2
< Sum of Squares of Sine and Cosine(Redirected from Difference of Squares of Cosecant and Cotangent)
Jump to navigation
Jump to search
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 $\sin x \ne 0$:
\(\ds \sin^2 x + \cos^2 x\) | \(=\) | \(\ds 1\) | Sum of Squares of Sine and Cosine | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 1 + \frac {\cos^2 x} {\sin^2 x}\) | \(=\) | \(\ds \frac 1 {\sin^2 x}\) | 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 |
$\blacksquare$
Also presented as
This result can also be presented 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
- 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: $4$
- 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