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: Simplify \( \cot^2 \left\{ \csc^{-1}(3) \right\} \)
Let \( \csc^{-1}(3) = \theta \). Then: \[ \csc \theta = 3 \implies \sin \theta = \frac{1}{3}. \] Using the identity \( \cot^2 \theta = \csc^2 \theta - 1 \): \[ \cot^2 \theta = \csc^2 \theta - 1 = 3^2 - 1 = 9 - 1 = 8. \] 
Step 2: Simplify \( \sin^2 \left\{ \cos^{-1} \left( \frac{1}{3} \right) \right\} \)
Let \( \cos^{-1} \left( \frac{1}{3} \right) = \phi \). Then: \[ \cos \phi = \frac{1}{3}. \] Using the identity \( \sin^2 \phi = 1 - \cos^2 \phi \): \[ \sin^2 \phi = 1 - \left( \frac{1}{3} \right)^2 = 1 - \frac{1}{9} = \frac{8}{9}. \] 
Step 3: Combine the results
\[ \cot^2 \left\{ \csc^{-1}(3) \right\} + \sin^2 \left\{ \cos^{-1} \left( \frac{1}{3} \right) \right\} = 8 + \frac{8}{9}. \] Simplify: \[ 8 + \frac{8}{9} = \frac{72}{9} + \frac{8}{9} = \frac{80}{9}. \] 
Conclusion: The value is \( \frac{80}{9} \).