Question

If \( \sin(x + y) + \cos(x + y) = \sin \left[ \cos^{-1} \left( \frac{1}{3} \right) \right] \), then \[ \frac{dy}{dx} = \]

• A. \( \frac{1}{2} \)
• B. \( -1 \)
• C. 1
• D. 0

Hint:

When differentiating trigonometric expressions, always remember to apply the chain rule and simplify the resulting expressions.

Answer 1

The Correct Option is B

Step 1: Use the given equation.
We are given the equation \( \sin(x + y) + \cos(x + y) = \sin \left[ \cos^{-1} \left( \frac{1}{3} \right) \right] \). First, we need to simplify this expression. Using the identity for \( \cos^{-1} \left( \frac{1}{3} \right) \), we can calculate the angle whose cosine is \( \frac{1}{3} \), which will help in further steps.

Step 2: Differentiate the equation.
Next, we differentiate both sides of the equation with respect to \( x \). This involves applying the chain rule for trigonometric functions. After performing the differentiation, we find that \( \frac{dy}{dx} = -1 \).

Step 3: Conclusion.
Thus, the correct answer is \( -1 \), corresponding to option (B).