Question

If \( \cos(\theta + \phi) = \frac{3}{5} \) and \( \sin(\theta - \phi) = \frac{5}{13} \), \( 0<\theta, \phi<\frac{\pi}{4} \), then \( \cot(2\theta) \) has the value:

• A. \( \frac{16}{63} \)
• B. \( \frac{63}{16} \)
• C. \( \frac{3}{13} \)
• D. \( \frac{13}{3} \)

Hint:

Remember the signs of trigonometric functions in different quadrants and use compound angle formulas correctly.

Answer 1

The Correct Option is A

Step 1: Find \( \sin(\theta + \phi) \) and \( \cos(\theta - \phi) \).
Given \( \cos(\theta + \phi) = \frac{3}{5} \) and \( 0<\theta + \phi<\frac{\pi}{2} \), \[ \sin(\theta + \phi) = \sqrt{1 - \left(\frac{3}{5}\right)^2} = \frac{4}{5} \] Given \( \sin(\theta - \phi) = \frac{5}{13} \) and \( -\frac{\pi}{4}<\theta - \phi<\frac{\pi}{4} \), \[ \cos(\theta - \phi) = \sqrt{1 - \left(\frac{5}{13}\right)^2} = \frac{12}{13} \]
Step 2: Use sum formula for \( \sin(2\theta) \) and \( \cos(2\theta) \).
Since \( 2\theta = (\theta + \phi) + (\theta - \phi) \), \[ \sin(2\theta) = \sin(\theta + \phi)\cos(\theta - \phi) + \cos(\theta + \phi)\sin(\theta - \phi) = \left(\frac{4}{5}\right)\left(\frac{12}{13}\right) + \left(\frac{3}{5}\right)\left(\frac{5}{13}\right) = \frac{48 + 15}{65} = \frac{63}{65} \] \[ \cos(2\theta) = \cos(\theta + \phi)\cos(\theta - \phi) - \sin(\theta + \phi)\sin(\theta - \phi) = \left(\frac{3}{5}\right)\left(\frac{12}{13}\right) - \left(\frac{4}{5}\right)\left(\frac{5}{13}\right) = \frac{36 - 20}{65} = \frac{16}{65} \]
Step 3: Calculate \( \cot(2\theta) \). \[ \cot(2\theta) = \frac{\cos(2\theta)}{\sin(2\theta)} = \frac{\frac{16}{65}}{\frac{63}{65}} = \frac{16}{63} \]