If $\sec heta \cosh y=\operatorname{cosec} x$ and $\operatorname{cosec} heta \sinh y=\sec x$, then $\sinh ^2 y=$

Question

If $\sec 	heta \cosh y=\operatorname{cosec} x$ and $\operatorname{cosec} 	heta \sinh y=\sec x$, then $\sinh ^2 y=$

Aicharya · Accepted Answer

Since, $\sec 	heta \cosh y=\operatorname{cosec} x$ and $\operatorname{cosec} 	heta \sinh y=\sec x$ If we square and add Eqs. (i) and (ii), $ \begin{array}{cc} \Rightarrow & 1=\sin ^2 x \cosh ^2 y+\cos ^2 x \sinh ^2 y \ \Rightarrow & \sin ^2 x\left(1+\sinh ^2 yight)+\cos ^2 x \sinh ^2 y=1 \ \Rightarrow & \sinh ^2 y+\sin ^2 x=1 \ \Rightarrow & \sinh ^2 y=1-\sin ^2 x=\cos ^2 x \end{array} $

If $\sec \theta \cosh y=\operatorname{cosec} x$ and $\operatorname{cosec} \theta \sinh y=\sec x$, then…

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