Question

The value of \(\tan ^2\left(\sec ^{-1} 4\right)+\cot ^2\left(\operatorname{cosec}^{-1} 3\right)\) is

• A. 15
• B. 25
• C. 23
• D. 7

Hint:

Use tan^2 = sec^2 - 1 and cot^2 = cosec^2 - 1.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
Let \(\theta = \sec^{-1}4\), so \(\sec\theta = 4\), and \(\phi = \operatorname{cosec}^{-1}3\), so \(\operatorname{cosec}\phi = 3\).

Step 2: First term
\(\tan^2\theta = \sec^2\theta - 1 = 16 - 1 = 15\).

Step 3: Second term
\(\cot^2\phi = \operatorname{cosec}^2\phi - 1 = 9 - 1 = 8\).

Step 4: Add
\(15 + 8 = 23\).

Final Answer:
The value is 23. \[ \boxed{\text{(C) }23} \]