If $\sin x \cosh y=\cos heta$ and $\cos x \sinh y=\sin heta$, then $\sin ^2 x+\cosh ^2 y=$

Question

If $\sin x \cosh y=\cos 	heta$ and $\cos x \sinh y=\sin 	heta$, then $\sin ^2 x+\cosh ^2 y=$

Aicharya · Accepted Answer

Given $\cos 	heta=\sin x \cdot \cos h y$ $\sin 	heta=\cos x \cdot \sin h y$ Now $\cos ^2 	heta+\sin ^2 	heta=(\sin x \cosh y)^2+(\cos x \sin h y)^2$ $ \begin{aligned} & \Rightarrow \quad 1=\sin ^2 x \cos h^2 y+\left(1-\sin ^2 xight) \sinh ^2 y \ & \Rightarrow \quad 1=\sin ^2 x\left(\cosh ^2 y-\sin h^2 yight)+\sin h^2 y \ & \Rightarrow \quad 1=\sin ^2 x(1)+\sin ^2 y \quad\left\{\because \cos h^2 y-\sin h^2 y=1ight. \ & \Rightarrow \quad 1+1=\sin ^2 x+\left(1+\sin h^2 yight) \ & \Rightarrow \quad 2=\sin ^2 x+\cosh ^2 y \quad\left\{\because 1+\sin h^2 y=\cosh ^2 yight. \end{aligned} $

If $\sin x \cosh y=\cos \theta$ and $\cos x \sinh y=\sin \theta$, then $\sin ^2 x+\cosh ^2 y=$

Solution