Question
Evaluate:
\[ \cot\left(\frac{\pi}{4} – 2\cot^{-1}(3)\right) \]
Solution
Let
\[ \theta = \cot^{-1}(3) \Rightarrow \cot\theta = 3 \Rightarrow \tan\theta = \frac{1}{3} \]
Find tan(2θ):
\[ \tan 2\theta = \frac{2\tan\theta}{1 – \tan^2\theta} = \frac{2 \cdot \frac{1}{3}}{1 – \frac{1}{9}} = \frac{2/3}{8/9} = \frac{3}{4} \]
Now use identity:
\[ \cot\left(\frac{\pi}{4} – 2\theta\right) = \frac{1 + \tan 2\theta}{1 – \tan 2\theta} \]
\[ = \frac{1 + \frac{3}{4}}{1 – \frac{3}{4}} = \frac{7/4}{1/4} = 7 \]
Final Answer:
\[ \boxed{7} \]
Key Concept
Convert cot⁻¹ into tan form and use double-angle identities.