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: Understanding the Concept:
Let \(\cot\theta=\dfrac{1}{\sqrt{x^{2}-1}}\), so \(\tan\theta=\sqrt{x^{2}-1}\).

Step 2: Building a right triangle:
With opposite \(=\sqrt{x^2-1}\) and adjacent \(=1\), the hypotenuse is \(\sqrt{1+(x^2-1)}=x\).

Step 3: Reading off sec theta:
\(\sec\theta=\dfrac{\text{hyp}}{\text{adj}}=\dfrac{x}{1}=x\), so \(\theta=\sec^{-1}x\).

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