Question

The value of \( \sec^2(\tan^{-1} 3) + \csc^2(\cot^{-1} 2) \) is equal to:

• A. \( 5 \)
• B. \( 13 \)
• C. \( 15 \)
• D. \( 23 \)
• E. \( 25 \)

Hint:

Don't bother converting the inverse tan into a secant using triangles unless you have to. Using the $1+\tan^2$ identity is almost always faster.

Answer 1

The Correct Option is C

Understanding the Concept: Use the trigonometric identities: \[ \sec^2 \theta = 1 + \tan^2 \theta \] \[ \csc^2 \theta = 1 + \cot^2 \theta \]

Step 1: Apply identities to the terms.
Let \( \alpha = \tan^{-1} 3 \) and \( \beta = \cot^{-1} 2 \). The expression is \( \sec^2 \alpha + \csc^2 \beta \). Substituting the identities: \[ (1 + \tan^2 \alpha) + (1 + \cot^2 \beta) \]

Step 2: Evaluate the trig functions of inverse trig functions.
Since \( \alpha = \tan^{-1} 3 \), then \( \tan \alpha = 3 \). Since \( \beta = \cot^{-1} 2 \), then \( \cot \beta = 2 \).

Step 3: Plug in the values.
\[ [1 + (3)^2] + [1 + (2)^2] \] \[ [1 + 9] + [1 + 4] \] \[ 10 + 5 = 15 \]