Question

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

• A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
• B. Both Assertion (A) and Reason (R) are true, but Reason (R) is it{not} the correct explanation of the Assertion (A).
• C. Assertion (A) is true, but Reason (R) is false.
• D. Assertion (A) is false, but Reason (R) is true.

Hint:

Use trigonometric identities and triangles to simplify expressions involving inverse trigonometric functions.

Answer 1

The Correct Option is C

1. Evaluate \( \sec^2 \left( \tan^{-1} \frac{1}{2} \right) \): Let \( \theta = \tan^{-1} \frac{1}{2} \), so \( \tan \theta = \frac{1}{2} \). Construct a right triangle: \[ {Opposite} = 1, \quad {Adjacent} = 2, \quad {Hypotenuse} = \sqrt{1^2 + 2^2} = \sqrt{5}. \] Then: \[ \sec^2 \theta = \frac{ {Hypotenuse}^2}{ {Adjacent}^2} = \frac{5}{4}. \] 2. Evaluate \( \csc^2 \left( \cot^{-1} \frac{1}{3} \right) \): Let \( \phi = \cot^{-1} \frac{1}{3} \), so \( \cot \phi = \frac{1}{3} \). Construct a right triangle: \[ {Adjacent} = 1, \quad {Opposite} = 3, \quad {Hypotenuse} = \sqrt{1^2 + 3^2} = \sqrt{10}. \] Then: \[ \csc^2 \phi = \frac{ {Hypotenuse}^2}{ {Opposite}^2} = \frac{10}{9}. \] 3. Add the results: \[ \sec^2 \left( \tan^{-1} \frac{1}{2} \right) + \csc^2 \left( \cot^{-1} \frac{1}{3} \right) = \frac{5}{4} + \frac{10}{9}. \] Taking the LCM: \[ \frac{5}{4} + \frac{10}{9} = \frac{45 + 40}{36} = \frac{85}{36}. \]
Final Answer: \( \boxed{\frac{85}{36}} \)