Sum of Squares of Sine and Cosine/Corollary 1
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Corollary to Sum of Squares of Sine and Cosine
For all $x \in \C$:
- $\sec^2 x - \tan^2 x = 1 \quad \text {(when $\cos x \ne 0$)}$
where $\sec$, $\tan$ and $\cos$ are secant, tangent and cosine respectively.
Proof
When $\cos x \ne 0$:
\(\ds \cos^2 x + \sin^2 x\) | \(=\) | \(\ds 1\) | Sum of Squares of Sine and Cosine | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 1 + \frac {\sin^2 x} {\cos^2 x}\) | \(=\) | \(\ds \frac 1 {\cos^2 x}\) | dividing both sides by $\cos^2 x$, as $\cos x \ne 0$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds 1 + \tan^2 x\) | \(=\) | \(\ds \sec^2 x\) | Definition of Tangent Function and Definition of Secant Function | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sec^2 x - \tan^2 x\) | \(=\) | \(\ds 1\) | rearranging |
$\blacksquare$
Also presented as
Difference of Squares of Secant and Tangent can also be presented as:
- $\sec^2 x = 1 + \tan^2 x \quad \text {(when $\cos x \ne 0$)}$
or:
- $\tan^2 x = \sec^2 x - 1 \quad \text {(when $\cos x \ne 0$)}$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: Formulae $(2)$
- 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: $5.20$
- 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