Evaluate:

Question:

Evaluate: $\sin \left\{\cos ^{-1}\left(-\frac{3}{5}\right)+\cot ^{-1}\left(-\frac{5}{12}\right)\right\}$

Solution:

$\sin \left\{\cos ^{-1}\left(-\frac{3}{5}\right)+\cot ^{-1}\left(-\frac{5}{12}\right)\right\}=\sin \left\{\pi-\cos ^{-1}\left(\frac{3}{5}\right)+\pi-\cot ^{-1}\left(\frac{5}{12}\right)\right\}$

$=\sin \left\{2 \pi-\left[\cos ^{-1}\left(\frac{3}{5}\right)+\cot ^{-1}\left(\frac{5}{12}\right)\right]\right\}$

$=-\sin \left\{\cos ^{-1}\left(\frac{3}{5}\right)+\cot ^{-1}\left(\frac{5}{12}\right)\right\}$

$=-\sin \left\{\sin ^{-1}\left[\sqrt{1-\left(\frac{3}{5}\right)^{2}}\right]+\sin ^{-1}\left[\frac{\frac{12}{5}}{\sqrt{1+\left(\frac{12}{5}\right)^{2}}}\right]\right\}$

$=-\sin \left(\sin ^{-1} \frac{4}{5}+\sin ^{-1} \frac{12}{13}\right)$

$=-\sin \left\{\sin ^{-1}\left[\frac{4}{5} \times \sqrt{1-\left(\frac{12}{13}\right)^{2}}+\frac{12}{13} \times \sqrt{1-\left(\frac{4}{5}\right)^{2}}\right]\right\}$

$=-\sin \left[\sin ^{-1}\left(\frac{20}{65}+\frac{36}{65}\right)\right]$

$=-\sin \left[\sin ^{-1}\left(\frac{56}{65}\right)\right]$

$=-\frac{56}{65}$

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