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: Use right triangles:
For $\tan\theta = 3$: opposite 3, adjacent 1, hypotenuse $\sqrt{10}$, so $\sec\theta = \sqrt{10}$ and $\sec^2\theta = 10$.
For $\sec\phi = 3$: hypotenuse 3, adjacent 1, opposite $\sqrt{8}$, so $\tan\phi = \sqrt8$ and $\tan^2\phi = 8$.

Step 2: Subtract:
$10 - 8 = 2$.

Step 3: Note:
Both triangles use the Pythagorean theorem, so the identity route and this route agree.

Final Answer:
2, option (C). \[ \boxed{2 \text{ (C)}} \]