Problem
Simplify: \( \cot^{-1}\left(\frac{a}{\sqrt{x^2 – a^2}}\right), \quad |x| > a \)
Solution
Let:
\[ \theta = \cot^{-1}\left(\frac{a}{\sqrt{x^2 – a^2}}\right) \]
Then,
\[ \cot \theta = \frac{a}{\sqrt{x^2 – a^2}} \]
Taking reciprocal:
\[ \tan \theta = \frac{\sqrt{x^2 – a^2}}{a} \]
Now, construct a right triangle:
- Opposite = \( \sqrt{x^2 – a^2} \)
- Adjacent = \( a \)
- Hypotenuse = \( x \)
Thus,
\[ \sin \theta = \frac{\sqrt{x^2 – a^2}}{x}, \quad \cos \theta = \frac{a}{x} \]
Hence,
\[ \theta = \cos^{-1}\left(\frac{a}{x}\right) \]
Final Answer
\[ \boxed{\cos^{-1}\left(\frac{a}{x}\right)} \]