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

• Correct answer
  A

  $\cos ^2 x$
• B

  $\cos x$
• C

  $\sin ^2 x$
• D

  $\sin x$

Solution

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

Hence, the correct answer is option: 'A'.