Question

Write \(\cot^{-1}\!\Big(\dfrac{1}{\sqrt{x^{2}-1}}\Big)\), \(x>1\) in the simplest form.

Hint:

Set cot(theta) equal to the given ratio and build a right triangle to identify sec(theta).

Answer 1

Step 1: Alternative substitution x = sec(phi):
Let \(x=\sec\phi\) with \(0<\phi<\pi/2\) (valid since \(x>1\)). Then \(\sqrt{x^2-1}=\sqrt{\sec^2\phi-1}=\tan\phi\).

Step 2: Substituting into the expression:
\(\cot^{-1}\Big(\dfrac{1}{\tan\phi}\Big)=\cot^{-1}(\cot\phi)=\phi\).

Step 3: Expressing in terms of x:
Since \(x=\sec\phi\), \(\phi=\sec^{-1}x\).

Final Answer:
\[ \boxed{\sec^{-1}x} \]