Question

Find the value of the following expression: \[ \tan^2(\sec^{-1}4) + \cot(\csc^{-1}3) \]

• A. \( 1 \)
• B. \( 2 \)
• C. \( 3 \)
• D. \( 4 \)

Hint:

\textbf{Quick Tip:} For inverse trigonometric functions like \( \sec^{-1}x \) or \( \csc^{-1}x \), recall that the secant and cosecant functions are related to cosine and sine, respectively. Use Pythagorean identities to simplify the problem and find the required trigonometric values.

Answer 1

The Correct Option is B

Step 1: Solve \( \tan^2(\sec^{-1}4) \) Recall that \( \sec^{-1}x \) gives an angle whose secant is \( x \). Thus: \[ \sec(\theta) = 4 \quad \text{where} \quad \theta = \sec^{-1}4 \] Since \( \sec(\theta) = \frac{1}{\cos(\theta)} \), we have: \[ \cos(\theta) = \frac{1}{4} \] Now, use the Pythagorean identity to find \( \sin(\theta) \): \[ \sin^2(\theta) + \cos^2(\theta) = 1 \] \[ \sin^2(\theta) = 1 - \left( \frac{1}{4} \right)^2 = 1 - \frac{1}{16} = \frac{15}{16} \] \[ \sin(\theta) = \frac{\sqrt{15}}{4} \] Next, calculate \( \tan^2(\theta) \): \[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{\frac{\sqrt{15}}{4}}{\frac{1}{4}} = \sqrt{15} \] Thus: \[ \tan^2(\theta) = (\sqrt{15})^2 = 15 \] Step 2: Solve \( \cot(\csc^{-1}3) \) Recall that \( \csc^{-1}x \) gives an angle whose cosecant is \( x \). Thus: \[ \csc(\theta) = 3 \quad \text{where} \quad \theta = \csc^{-1}3 \] Since \( \csc(\theta) = \frac{1}{\sin(\theta)} \), we have: \[ \sin(\theta) = \frac{1}{3} \] Now, use the Pythagorean identity to find \( \cos(\theta) \): \[ \sin^2(\theta) + \cos^2(\theta) = 1 \] \[ \cos^2(\theta) = 1 - \left( \frac{1}{3} \right)^2 = 1 - \frac{1}{9} = \frac{8}{9} \] \[ \cos(\theta) = \frac{\sqrt{8}}{3} = \frac{2\sqrt{2}}{3} \] Next, calculate \( \cot(\theta) \): \[ \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} = \frac{\frac{2\sqrt{2}}{3}}{\frac{1}{3}} = 2\sqrt{2} \] Step 3: Add the results Now, add the results from Step 1 and Step 2: \[ \tan^2(\sec^{-1}4) + \cot(\csc^{-1}3) = 15 + 2\sqrt{2} \] Answer: The value of the expression is \( 15 + 2\sqrt{2} \). Therefore, the correct answer is option (2).