Question

The solution of the differential equation $ \frac{dy}{dx} \tan y = \sin(x + y) + \sin(x - y) $ is

• A. \( \sec x = -2 \sec y + C \)
• B. \( \sec y = 2 \cos y + C \)
• C. \( \sec y = -2 \cos x + C \)
• D. \( \sec x = -2 \cos y + C \)

Hint:

When dealing with trigonometric functions in differential equations, remember to apply sum-to-product identities for simplification, which can make the equation easier to solve.

Answer 1

The Correct Option is C

The given differential equation is: \[ \frac{dy}{dx} \tan y = \sin(x + y) + \sin(x - y) \] Using the sum-to-product identity for sine, we have: \[ \sin(x + y) + \sin(x - y) = 2 \sin x \cos y \] Thus, the equation becomes: \[ \frac{dy}{dx} \tan y = 2 \sin x \cos y \] By separating variables and integrating both sides, we get: \[ \sec y = -2 \cos x + C \] This is the solution to the given differential equation.