Question

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

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

Hint:

For mixed angle sums: \begin{itemize} \item Express \( 2\theta = (\theta+\phi)+(\theta-\phi) \). \item Then use sum formulas. \end{itemize}

Answer 1

The Correct Option is A

Concept: Use: \[ \cos(A+B), \quad \sin(A-B) \] and convert into sine/cosine of individual angles. Step 1: Convert ratios. \[ \cos(\theta+\phi)=\frac{3}{5} \Rightarrow \sin(\theta+\phi)=\frac{4}{5} \] \[ \sin(\theta-\phi)=\frac{5}{13} \Rightarrow \cos(\theta-\phi)=\frac{12}{13} \] Step 2: Use formulas. \[ \cos2\theta = \cos[(\theta+\phi)+(\theta-\phi)] \] \[ = \cos(\theta+\phi)\cos(\theta-\phi) - \sin(\theta+\phi)\sin(\theta-\phi) \] \[ = \frac{3}{5}\cdot\frac{12}{13} - \frac{4}{5}\cdot\frac{5}{13} \] \[ = \frac{36 - 20}{65} = \frac{16}{65} \] Step 3: Find sine. \[ \sin2\theta = \sin[(\theta+\phi)+(\theta-\phi)] \] \[ = \sin(\theta+\phi)\cos(\theta-\phi) + \cos(\theta+\phi)\sin(\theta-\phi) \] \[ = \frac{4}{5}\cdot\frac{12}{13} + \frac{3}{5}\cdot\frac{5}{13} \] \[ = \frac{48 + 15}{65} = \frac{63}{65} \] Step 4: Compute cotangent. \[ \cot(2\theta) = \frac{\cos2\theta}{\sin2\theta} = \frac{16}{63} \]