Question

Considering only the principal values of the inverse trigonometric functions, the value of
\(\tan(\sin^{-1}(\frac{3}{5})-2\cos^{-1}(\frac{2}{\sqrt5}))\)
is

• A. \(\frac{7}{24}\)
• B. \(\frac{-7}{24}\)
• C. \(\frac{-5}{24}\)
• D. \(\frac{5}{24}\)

Answer 1

The Correct Option is B

Let \[ 2 \cos^{-1} \left( \frac{2}{\sqrt{5}} \right) = \theta. \]

Step 1: Expressing in Terms of Half-Angle 

\[ \frac{2}{\sqrt{5}} = \cos \frac{\theta}{2} \]

Therefore, \[ \tan \frac{\theta}{2} = \frac{1}{2}. \]

Step 2: Calculating Individual Trigonometric Functions

\[ \tan \sin^{-1} \left( \frac{3}{5} \right) = \frac{3}{4}, \quad \tan \cos^{-1} \left( \frac{2}{\sqrt{5}} \right) = \frac{4}{3}. \]

Step 3: Applying the Tangent Subtraction Formula

\[ \tan \left( \sin^{-1} \frac{3}{5} - \cos^{-1} \frac{2}{\sqrt{5}} \right) = \frac{\frac{3}{4} - \frac{4}{3}}{1 + \frac{3}{4} \cdot \frac{4}{3}} \]

Final Calculation

\[ = \frac{\frac{9}{12} - \frac{16}{12}}{1 + \frac{12}{12}} \] \[ = \frac{-7}{24} \]