Reciprocal of One Plus Cosecant
Jump to navigation
Jump to search
Theorem
- $\dfrac {\cos^2 x + 3 \sin x - 1} {\cos^2 x + 2 \sin x + 2} = \dfrac 1 {1 + \csc x}$
Proof
\(\ds \frac {\cos^2 x + 3 \sin x - 1} {\cos^2 x + 2 \sin x + 2}\) | \(=\) | \(\ds \frac {1 - \sin^2 x + 3 \sin x - 1} {1 - \sin^2 x + 2 \sin x + 2}\) | Sum of Squares of Sine and Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sin^2 x - 3 \sin x} {\sin^2 x - 2 \sin x - 3}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sin x \paren {\sin x - 3} } {\paren {\sin x - 3} \paren {\sin x + 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sin x} {\sin x + 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {1 + \dfrac 1 {\sin x} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {1 + \csc x}\) | Definition of Real Cosecant Function |
$\blacksquare$