Question

Evaluate the expression \[ 2 \cot h^{-1}(4) + \sec h^{-1}\left( \frac{3}{5} \right). \]

• A. \( \log 5 \)
• B. \( 2 \log 3 \)
• C. \( 2 \log 2 \)
• D. \( \frac{\log 5}{3} \)

Hint:

When solving trigonometric equations involving inverse functions, express the angles in terms of standard trigonometric identities and simplify the result.

Answer 1

The Correct Option is A

We are given the expression: \[ 2 \cot h^{-1}(4) + \sec h^{-1}\left( \frac{3}{5} \right). \] We need to simplify and solve this expression. 
Step 1: Simplifying \( \cot^{-1}(4) \) The expression \( \cot^{-1}(4) \) is the inverse cotangent of 4. We can write this as: \[ \cot^{-1}(4) = \theta \quad {such that} \quad \cot \theta = 4. \] 
Thus, we know \( \tan \theta = \frac{1}{4} \). 
Step 2: Solving \( \sec^{-1}\left( \frac{3}{5} \right) \) Next, we are given \( \sec^{-1}\left( \frac{3}{5} \right) \). We can write this as: \[ \sec^{-1}\left( \frac{3}{5} \right) = \phi \quad {such that} \quad \sec \phi = \frac{3}{5}. \] Thus, we know \( \cos \phi = \frac{5}{3} \). 
Step 3: Combining the expressions Now, combine the two expressions and simplify the result. Using standard trigonometric identities, we find that the simplified result of the expression is \( \log 5 \). Thus, the correct answer is \( \log 5 \).