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

To simplify the expression \( \cos^{-1} \left( \frac{12}{13} \cos x + \frac{5}{13} \sin x \right) \) for \( \frac{\pi}{2} \leq x \leq \frac{3\pi}{4} \), we proceed as follows:

1. The expression inside the inverse cosine function can be rewritten using a trigonometric identity. Let \( a = \frac{12}{13} \) and \( b = \frac{5}{13} \).
2. Since \( \sqrt{a^2 + b^2} = \sqrt{\left(\frac{12}{13}\right)^2 + \left(\frac{5}{13}\right)^2} = 1 \), the expression can be transformed into the form \( \cos(\theta) \cos x + \sin(\theta) \sin x = \cos(x - \theta) \).
3. The angle \( \theta \) is defined by \( \theta = \tan^{-1}\left(\frac{b}{a}\right) = \tan^{-1}\left(\frac{5}{12}\right) \).
4. Thus, the original expression becomes \( \cos^{-1}(\cos(x - \theta)) \).
5. Given that \( \cos \) is non-increasing in the specified domain, \( \cos^{-1}(\cos(x - \theta)) = x - \theta \).
6. Substituting the value of \( \theta \), we get \( \cos^{-1} \left( \frac{12}{13} \cos x + \frac{5}{13} \sin x \right) = x - \tan^{-1}\left(\frac{5}{12}\right) \).

This result corresponds to the provided option.

Hence, the correct answer is
Option: \( x - \tan^{-1} \left(\frac{5}{12}\right) \).