Question

Evaluate: \[ \cot^2 \left\{ \csc^{-1}(3) \right\} + \sin^2 \left\{ \cos^{-1} \left( \frac{1}{3} \right) \right\}. \]

Hint:

For inverse trigonometric expressions, convert them to standard trigonometric forms using definitions, and simplify using identities.

Answer 1

Step 1: Evaluate \( \cot^2 \left\{ \csc^{-1}(3) \right \} \). Let \( \theta = \csc^{-1}(3) \), so \( \csc \theta = 3 \) and \( \sin \theta = \frac{1}{3} \). Using \( \cot^2 \theta = \csc^2 \theta - 1 \), we get \( \cot^2 \theta = 3^2 - 1 = 9 - 1 = 8 \).

Step 2: Evaluate \( \sin^2 \left\{ \cos^{-1} \left( \frac{1}{3} \right) \right \} \). Let \( \phi = \cos^{-1} \left( \frac{1}{3} \right) \), so \( \cos \phi = \frac{1}{3} \). Using \( \sin^2 \phi = 1 - \cos^2 \phi \), we get \( \sin^2 \phi = 1 - \left( \frac{1}{3} \right)^2 = 1 - \frac{1}{9} = \frac{8}{9} \).

Step 3: Combine the results. The sum is \( 8 + \frac{8}{9} = \frac{72}{9} + \frac{8}{9} = \frac{80}{9} \).

Conclusion: The final value is \( \frac{80}{9} \).