Question

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

• A. 1
• B. 2
• C. \( \sin 2\theta \)
• D. \( \cos 2\theta \)

Hint:

When working with hyperbolic functions, remember the identity \( \cosh^2 y - \sinh^2 y = 1 \), which can simplify calculations.

Answer 1

The Correct Option is B

We are given the following two equations: \[ \sin x \cosh y = \cos \theta \quad \text{and} \quad \cos x \sinh y = \sin \theta \] Step 1: Square both equations.
Squaring the first equation: \[ \left( \sin x \cosh y \right)^2 = \cos^2 \theta \] \[ \sin^2 x \cosh^2 y = \cos^2 \theta \quad \text{(Equation 1)} \] Squaring the second equation: \[ \left( \cos x \sinh y \right)^2 = \sin^2 \theta \] \[ \cos^2 x \sinh^2 y = \sin^2 \theta \quad \text{(Equation 2)} \] Step 2: Add the two equations.
Now add Equation 1 and Equation 2: \[ \sin^2 x \cosh^2 y + \cos^2 x \sinh^2 y = \cos^2 \theta + \sin^2 \theta \] Since \( \cos^2 \theta + \sin^2 \theta = 1 \), we get: \[ \sin^2 x \cosh^2 y + \cos^2 x \sinh^2 y = 1 \] Step 3: Use the identity for hyperbolic functions.
We know the identity: \[ \cosh^2 y - \sinh^2 y = 1 \] Thus, the expression \( \sin^2 x + \cosh^2 y \) is equal to 2.
Therefore, the value of \( \sin^2 x + \cosh^2 y \) is \( \boxed{2} \).