Cosecant of Complement equals Secant
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Theorem
- $\map \csc {\dfrac \pi 2 - \theta} = \sec \theta$ for $\theta \ne \paren {2 n + 1} \dfrac \pi 2$
where $\csc$ and $\sec$ are cosecant and secant respectively.
That is, the secant of an angle is the cosecant of its complement.
This relation is defined wherever $\cos \theta \ne 0$.
Proof
\(\ds \map \csc {\frac \pi 2 - \theta}\) | \(=\) | \(\ds \frac 1 {\map \sin {\frac \pi 2 - \theta} }\) | Cosecant is Reciprocal of Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {\cos \theta}\) | Sine of Complement equals Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \sec \theta\) | Secant is Reciprocal of Cosine |
The above is valid only where $\cos \theta \ne 0$, as otherwise $\dfrac 1 {\cos \theta}$ is undefined.
From Cosine of Half-Integer Multiple of Pi it follows that this happens when $\theta \ne \paren {2 n + 1} \dfrac \pi 2$.
$\blacksquare$
Also see
- Sine of Complement equals Cosine
- Cosine of Complement equals Sine
- Tangent of Complement equals Cotangent
- Cotangent of Complement equals Tangent
- Secant of Complement equals Cosecant
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: Functions of Angles in All Quadrants in terms of those in Quadrant I