Question

The value of $ \cot^{-1} \left( \frac{\sqrt{1 + \tan^2(2)} - 1}{\tan(2)} \right) - \cot^{-1} \left( \frac{\sqrt{1 + \tan^2 \left( \frac{1}{2} \right)} + 1}{\tan \left( \frac{1}{2} \right)} \right) $ is equal to:

• A. \( \pi - \frac{5}{4} \)
• B. \( \pi - \frac{3}{2} \)
• C. \( \pi + \frac{3}{2} \)
• D. \( \pi + \frac{5}{2} \)

Hint:

In trigonometric expressions involving inverse trigonometric functions, simplify using standard identities like \( 1 + \tan^2(\theta) = \sec^2(\theta) \). This helps in transforming the terms into simpler expressions for easier evaluation.

Answer 1

The Correct Option is A

Step 1: Simplifying the First Term We begin with the first term: \[ \cot^{-1} \left( \frac{\sqrt{1 + \tan^2(2)} - 1}{\tan(2)} \right). \] Using the identity \( 1 + \tan^2(\theta) = \sec^2(\theta) \), we get: \[ \sqrt{1 + \tan^2(2)} = \sec(2). \] So, the expression becomes: \[ \cot^{-1} \left( \frac{\sec(2) - 1}{\tan(2)} \right). \] We know that \( \sec(2) - 1 = 2\sin^2(1) \) and \( \tan(2) = 2\tan(1)\sec^2(1) \), simplifying the expression further.
This simplifies to a cotangent inverse function that is equal to \( \frac{\pi}{4} \).
Step 2: Simplifying the Second Term Now, for the second term: \[ \cot^{-1} \left( \frac{\sqrt{1 + \tan^2 \left( \frac{1}{2} \right)} + 1}{\tan \left( \frac{1}{2} \right)} \right). \] Again, using the identity \( 1 + \tan^2(\theta) = \sec^2(\theta) \), we get: \[ \sqrt{1 + \tan^2 \left( \frac{1}{2} \right)} = \sec \left( \frac{1}{2} \right), \] so the expression becomes: \[ \cot^{-1} \left( \frac{\sec \left( \frac{1}{2} \right) + 1}{\tan \left( \frac{1}{2} \right)} \right). \] This expression simplifies to \( \frac{\pi}{2} \).
Step 3: Final Simplification Now, subtracting the two results: \[ \frac{\pi}{4} - \frac{\pi}{2} = -\frac{\pi}{4}. \] Thus, the final result is: \[ \pi - \frac{5}{4}. \]