Question
If
\[ u = \cot^{-1}(\sqrt{\tan\theta}) – \tan^{-1}(\sqrt{\tan\theta}) \]
Find:
\[ \tan\left(\frac{\pi}{4} – \frac{u}{2}\right) \]
Solution
Let
\[ x = \sqrt{\tan\theta} \]
Then,
\[ u = \cot^{-1}x – \tan^{-1}x \]
Use identity:
\[ \cot^{-1}x = \frac{\pi}{2} – \tan^{-1}x \]
So,
\[ u = \left(\frac{\pi}{2} – \tan^{-1}x\right) – \tan^{-1}x = \frac{\pi}{2} – 2\tan^{-1}x \]
Thus,
\[ \frac{u}{2} = \frac{\pi}{4} – \tan^{-1}x \]
Now,
\[ \frac{\pi}{4} – \frac{u}{2} = \frac{\pi}{4} – \left(\frac{\pi}{4} – \tan^{-1}x\right) = \tan^{-1}x \]
Therefore,
\[ \tan\left(\frac{\pi}{4} – \frac{u}{2}\right) = \tan(\tan^{-1}x) = x \]
Substitute back:
\[ x = \sqrt{\tan\theta} \]
Final Answer:
\[ \boxed{\sqrt{\tan\theta}} \]
Key Concept
Convert cot⁻¹ into tan⁻¹ and simplify step-by-step using identities.