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Question Number 81471 by jagoll last updated on 13/Feb/20

$$\mathrm{prove}\mathrm{that}$$$$\tan \left(\mathrm{x}\right)=\sinh \left(\mathrm{y}\right)\mathrm{if}\sin \left(\mathrm{x}\right)=$$$$\tanh \left(\mathrm{y}\right).$$

Commented by john santu last updated on 13/Feb/20

$$\Rightarrow \sin \left(\mathrm{x}\right)=\tanh \left(\mathrm{y}\right)$$$$\mathrm{squaring}\Rightarrow \sin {}^2\left(\mathrm{x}\right)=\tanh {}^2\left(\mathrm{h}\right)$$$$1-\sin {}^2\left(\mathrm{x}\right)=1-\tanh {}^2\left(\mathrm{y}\right)$$$$\cos {}^2\left(\mathrm{x}\right)=\mathrm{sech}{}^2\left(\mathrm{y}\right)$$$$\cos \left(\mathrm{x}\right)=\mathrm{sech}\left(\mathrm{y}\right)$$$$\mathrm{we}\mathrm{get}\frac{\sin \left(\mathrm{x}\right)}{\cos \left(\mathrm{x}\right)}=\frac{\tanh \left(\mathrm{y}\right)}{\mathrm{sech}\left(\mathrm{y}\right)}=\sinh \left(\mathrm{y}\right)$$

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