## Inverse Trigonometric Equations

To solve inverse trigonometric equations use the following important identities:
$sin^{-1} x = \frac{\pi}{2} - cos^{-1} x$

$tan^{-1} x = \frac{\pi}{2} - cot^{-1} x$

Example:
$sin^{-1} \theta = cos^{-1} \theta$

$sin^{-1} \theta = cos^{-1} \theta = \frac{\pi}{2} - sin^{-1} \theta$

$2sin^{-1} \theta = \frac{\pi}{2}$

$sin^{-1} \theta = \frac{\pi}{4}$

$\theta = sin(\frac{\pi}{4})= \frac{\sqrt{2}}{2}$

Example:
$5tan^{-1} \theta = 4 cot^{-1} \theta$

$5tan^{-1} \theta = 4cot^{-1} \theta =4( \frac{\pi}{2} - tan^{-1} \theta )=2 \pi - 4 tan^{-1} \theta$

$9tan^{-1} \theta = 2 \pi$

$tan^{-1} \theta = \frac{2 \pi}{9}$

$\theta = tan(\frac{2 \pi}{9})$