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Q. $sin\left\{2\,cos^{-1}\left(- \frac{3}{5}\right)\right\}$ is equal to

Inverse Trigonometric Functions

Solution:

$sin\left\{2\,cos^{-1}\left(- \frac{3}{5}\right)\right\}$
$= 2\,sin\left\{cos^{-1}\left(- \frac{3}{5}\right)\right\}cos\left\{cos^{-1}\left(- \frac{3}{5}\right)\right\}$
$= 2\,sin\left\{\pi - cos^{-1} \frac{3}{5}\right\}\times \left(-\frac{3}{5}\right) = -\frac{6}{5}sin\left\{cos^{-1} \frac{3}{5}\right\}$
$= -\frac{6}{5}sin\left(sin^{-1}\sqrt{1-\frac{9}{25}}\right)$
$ = -\frac{6}{5} \times \frac{4}{5} = -\frac{24}{25}$