Question

\(sec^2(tan^{-1}3)-tan^2(sec^{-1}3) =\)

• A. \(0\)
• B. \(1\)
• C. \(2\)
• D. \(3\)

Hint:

Use sec^2 theta = 1 + tan^2 theta, so each term can be found directly.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Both terms are squares of trigonometric functions of an inverse function. We can use the identity \(\sec^2\theta = 1 + \tan^2\theta\).

Step 2: Key Formula or Approach:
1. If \(\theta = \tan^{-1}3\), then \(\tan\theta = 3\).
2. If \(\phi = \sec^{-1}3\), then \(\sec\phi = 3\).

Step 3: Detailed Explanation:
First term:
\[ \sec^2(\tan^{-1}3) = 1 + \tan^2(\tan^{-1}3) = 1 + 9 = 10 \]
Second term:
\[ \tan^2(\sec^{-1}3) = \sec^2(\sec^{-1}3) - 1 = 9 - 1 = 8 \]
Subtract:
\[ 10 - 8 = 2 \]
Options (A), (B) and (D) come from slips such as using \(\tan^2 = \sec^2\) or forgetting to square the 3.

Final Answer:
The value of the expression is 2, option (C). \[ \boxed{2 \text{ (C)}} \]