Question

Find \[ \sec^2 \left( \tan^{-1} 2 \right) + \csc^2 \left( \cot^{-1} 3 \right) = ? \]

• A. 5
• B. 15
• C. 10
• D. 1

Hint:

When solving trigonometric identities involving inverse functions, remember the Pythagorean identity \( \sec^2 \theta = 1 + \tan^2 \theta \) and \( \csc^2 \theta = 1 + \cot^2 \theta \).

Answer 1

The Correct Option is B

Given: \[ \sec^2 \left( \tan^{-1} 2 \right) + \csc^2 \left( \cot^{-1} 3 \right) \] 1. First, for \( \sec^2 \left( \tan^{-1} 2 \right) \), we know that: \[ \tan \theta = 2 \quad \Rightarrow \quad \sec^2 \theta = 1 + \tan^2 \theta = 1 + 2^2 = 5 \] 2. For \( \csc^2 \left( \cot^{-1} 3 \right) \), we know that: \[ \cot \theta = 3 \quad \Rightarrow \quad \csc^2 \theta = 1 + \cot^2 \theta = 1 + 3^2 = 10 \] Therefore: \[ \sec^2 \left( \tan^{-1} 2 \right) + \csc^2 \left( \cot^{-1} 3 \right) = 5 + 10 = 15 \]