Question

If \( \frac{\pi}{2} \leq x \leq \frac{3\pi}{4} \), then \( \cos^{-1} \left( \frac{12}{13} \cos x + \frac{5}{13} \sin x \right) \) is equal to:

• A. \( x - \tan^{-1} \left(\frac{4}{3}\right) \)
• B. \( x - \tan^{-1} \left(\frac{5}{12}\right) \)
• C. \( x + \tan^{-1} \left(\frac{4}{5}\right) \)
• D. \( x + \tan^{-1} \left(\frac{5}{12}\right) \)

Hint:

To solve inverse trigonometric functions, use known identities such as \( \cos^{-1}(\cos \theta) = \theta \), ensuring that the angle is within the correct range.

Answer 1

The Correct Option is B

We are given: \[ \cos^{-1} \left( \frac{12}{13} \cos x + \frac{5}{13} \sin x \right). \] Using the identity for the sum of cosines: \[ \cos^{-1} \left( \cos \alpha \cos x + \sin \alpha \sin x \right) = \cos^{-1} \left( \cos (x - \alpha) \right). \] This implies: \[ x - \alpha \quad \text{because} \quad x - \alpha \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right). \] Thus, we have: \[ x - \tan^{-1} \left( \frac{5}{12} \right). \]