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Question Number 114122 by bemath last updated on 17/Sep/20

$$findthevalue\sin ({\cos }^{-1}\left(\frac{3}{5}\right)+{\tan }^{-1}\left(\frac{7}{13}\right))$$

Answered by mr W last updated on 17/Sep/20

$${\tan }^{-1}\frac{7}{13}={\sin }^{-1}\frac{7}{\sqrt{218}}={\cos }^{-1}\frac{13}{\sqrt{218}}$$$${\cos }^{-1}\frac{3}{5}={\sin }^{-1}\frac{4}{5}$$$$\sin ({\cos }^{-1}\left(\frac{3}{5}\right)+{\tan }^{-1}\left(\frac{7}{13}\right))$$$$=\left(\sin {\cos }^{-1}\frac{3}{5}\right)\left(\cos {\tan }^{-1}\frac{7}{13}\right)+\cos \left({\cos }^{-1}\frac{3}{5}\right)\sin \left({\tan }^{-1}\frac{7}{13}\right)$$$$=\left(\sin {\sin }^{-1}\frac{4}{5}\right)\left(\cos {\cos }^{-1}\frac{13}{\sqrt{218}}\right)+\cos \left({\cos }^{-1}\frac{3}{5}\right)\sin \left({\sin }^{-1}\frac{7}{\sqrt{218}}\right)$$$$=\frac{4}{5}\times \frac{13}{\sqrt{218}}+\frac{3}{5}\times \frac{7}{\sqrt{218}}$$$$=\frac{73}{5\sqrt{218}}$$

Commented by bemath last updated on 17/Sep/20

$$santuyy$$

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