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

prove that tan (x) = sinh (y) if sin (x)= tanh (y).

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

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

⇒sin (x)=tanh (y) squaring ⇒sin^2 (x)=tanh^2 (h) 1−sin^2 (x)=1−tanh^2 (y) cos^2 (x)= sech^2 (y) cos (x)=sech (y) we get ((sin (x))/(cos (x))) = ((tanh (y))/(sech (y))) = sinh (y)

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

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