Question

If \( 2 \cot^{-1} \left( \frac{4}{3} \right) = \cos^{-1} \left( \frac{x}{5} \right) \), find \( x \).

• A. 3
• B. 4
• C. 5
• D. 6

Hint:

When dealing with inverse trigonometric identities, use the basic trigonometric identities like \( \cos^2 \theta + \sin^2 \theta = 1 \) and the double angle formulas for efficient calculation.

Answer 1

The Correct Option is B

Step 1: Use inverse trigonometric identities.
We start with the equation: \[ 2 \cot^{-1} \left( \frac{4}{3} \right) = \cos^{-1} \left( \frac{x}{5} \right) \] Let \( \theta = \cot^{-1} \left( \frac{4}{3} \right) \), then: \[ \cot \theta = \frac{4}{3} \] Using the identity \( \cot \theta = \frac{\cos \theta}{\sin \theta} \), we know that: \[ \frac{\cos \theta}{\sin \theta} = \frac{4}{3} \] Thus, \( \cos \theta = 4k \) and \( \sin \theta = 3k \), where \( k \) is a constant. Since \( \cos^2 \theta + \sin^2 \theta = 1 \), we get: \[ (4k)^2 + (3k)^2 = 1 \] \[ 16k^2 + 9k^2 = 1 \] \[ 25k^2 = 1 \quad \Rightarrow \quad k = \frac{1}{5} \] Therefore, \( \cos \theta = \frac{4}{5} \) and \( \sin \theta = \frac{3}{5} \).
Step 2: Solve for \( x \).
Now, using the given equation, we have: \[ 2\theta = \cos^{-1} \left( \frac{x}{5} \right) \] So, \( \cos 2\theta = \frac{x}{5} \). Using the double angle identity for cosine: \[ \cos 2\theta = 2\cos^2 \theta - 1 \] Substitute \( \cos \theta = \frac{4}{5} \): \[ \cos 2\theta = 2 \left( \frac{4}{5} \right)^2 - 1 = 2 \times \frac{16}{25} - 1 = \frac{32}{25} - 1 = \frac{7}{25} \] Thus: \[ \frac{x}{5} = \frac{7}{25} \quad \Rightarrow \quad x = 4 \]
Final Answer: 4.