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Question Number 133976 by liberty last updated on 26/Feb/21

$${\sin }^{-1}\left(\frac{3}{5}\right)+{\tan }^{-1}\left(\frac{1}{7}\right)=?$$

Answered by bemath last updated on 26/Feb/21

$$\mathrm{let}{\sin }^{-1}\left(\frac{3}{5}\right)=\vartheta \Rightarrow \{\begin{array}{l}\sin \vartheta =\frac{3}{5}\\ \tan \vartheta =\frac{3}{4}\end{array}$$$$\Rightarrow {\sin }^{-1}\left(\frac{3}{5}\right)={\tan }^{-1}\left(\frac{3}{4}\right)$$$$$$$$\iff {\sin }^{-1}\left(\frac{3}{5}\right)+{\tan }^{-1}\left(\frac{1}{7}\right)=$$$${\tan }^{-1}\left(\frac{3}{4}\right)+{\tan }^{-1}\left(\frac{1}{7}\right)=$$$${\tan }^{-1}\left(\frac{\frac{3}{4}+\frac{1}{7}}{1-\frac{3}{4}.\frac{1}{7}}\right)={\tan }^{-1}\left(\frac{25}{25}\right)=\frac{\pi }{4}$$

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