Question
Simplify:
\[ \sin\left[\cot^{-1}\{\tan(\cos^{-1}x)\}\right] \]
Solution
Let
\[ \cos^{-1}x = \theta \Rightarrow \cos\theta = x \]
Then,
\[ \tan(\cos^{-1}x) = \tan\theta \]
Now,
\[ \sin\left(\cot^{-1}(\tan\theta)\right) \]
Let
\[ \cot^{-1}(\tan\theta) = \phi \Rightarrow \cot\phi = \tan\theta \Rightarrow \tan\phi = \cot\theta \]
So,
\[ \phi = \frac{\pi}{2} – \theta \]
Thus,
\[ \sin\phi = \sin\left(\frac{\pi}{2} – \theta\right) = \cos\theta \]
But \( \cos\theta = x \),
\[ \sin\left[\cot^{-1}\{\tan(\cos^{-1}x)\}\right] = x \]
Final Answer:
\[ \boxed{x} \]
Key Concept
Convert everything into a single variable and use complementary angle identities.