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Question: If \( \sin x\cosh y = \cos \theta \) and \( \cos x\sinh y = \sin \theta \) then \( {\sinh ^2}y = \) ...

If sin⁡xcosh⁡y=cos⁡θ\sin x\cosh y = \cos \theta and cos⁡xsinh⁡y=sin⁡θ\cos x\sinh y = \sin \theta then sinh⁡2y={\sinh ^2}y =
1)sin⁡2x 2)cosh⁡2x 3)cos⁡2x 4)1  1){\sin ^2}x \\\ 2){\cosh ^2}x \\\ 3){\cos ^2}x \\\ 4)1 \\\

Explanation

Solution

Hint : Here first of all we will convert the given expressions in squares as the resultant answer we required is in square form. Later we will simplify the given two expressions in the form of cosine and then will simplify for the required value.

Complete step-by-step answer :
Take the given two expressions:
sin⁡xcosh⁡y=cos⁡θ\sin x\cosh y = \cos \theta …. (A)
cos⁡xsinh⁡y=sin⁡θ\cos x\sinh y = \sin \theta …. (B)
Take the square of both the above equations and add them to the other.
sin⁡2xcosh⁡2y+cos⁡2xsinh⁡2y=cos⁡2θ+sin⁡2θ{\sin ^2}x{\cosh ^2}y + {\cos ^2}x{\sinh ^2}y = {\cos ^2}\theta + {\sin ^2}\theta
Simplify the above expression by using the identity cos⁡2θ+sin⁡2θ=1{\cos ^2}\theta + {\sin ^2}\theta = 1
sin⁡2xcosh⁡2y+cos⁡2xsinh⁡2y=1{\sin ^2}x{\cosh ^2}y + {\cos ^2}x{\sinh ^2}y = 1
Convert the above expression in the form of cosine by using the identity that sin⁡2x=1−cos⁡2x{\sin ^2}x = 1 - {\cos ^2}x and sinh⁡2x=cosh⁡2x−1{\sinh ^2}x = {\cosh ^2}x - 1
(1−cos⁡2x)cosh⁡2y+cos⁡2x(cosh⁡2y−1)=1(1 - {\cos ^2}x){\cosh ^2}y + {\cos ^2}x({\cosh ^2}y - 1) = 1
Simplify the above expression by finding the product of the terms –
cosh⁡2y−cos⁡2xcosh⁡2y+cos⁡2xcosh⁡2y−cos⁡2x=1{\cosh ^2}y - {\cos ^2}x{\cosh ^2}y + {\cos ^2}x{\cosh ^2}y - {\cos ^2}x = 1
Like terms with the same value and opposite sign cancels each other.
cosh⁡2y−cos⁡2x=1{\cosh ^2}y - {\cos ^2}x = 1
Move terms to get framed the above expression in the form of the identity. When you move any terms from one side to the opposite side then the sign of the terms also changes. Positive term becomes negative and the negative term becomes positive.
cosh⁡2y−1=cos⁡2x{\cosh ^2}y - 1 = {\cos ^2}x
By using the identity, above expression can be written as –
sinh⁡2y=cos⁡2x{\sinh ^2}y = {\cos ^2}x
From the given multiple choices, the first option is the correct answer.
So, the correct answer is “Option 1”.

Note : Be careful about the sign convention while moving any terms from one side to the opposite side. Sign of the term is always changed while moving any term from one side to the opposite side. Know the difference between the angle and the hyperbolic angle and apply its identity wisely. Know the identities for sine, cosine and its correlations.