Question

(e) If \( y = x \cos (a + y) \) and \( \cos a \neq \pm 1 \), prove that \[ \frac{dy}{dx} = \frac{\cos^2 (a + y)}{\sin a}. \]

Hint:

Implicit differentiation is essential when the dependent variable appears inside a trigonometric function.

Answer 1

Differentiating both sides with respect to \( x \): \[ \frac{dy}{dx} = \cos (a + y) - x \sin (a + y) \frac{dy}{dx}. \] Rearranging terms: \[ \frac{dy}{dx} = \frac{\cos (a + y)}{1 + x \sin (a + y)}. \]