Question

Considering only the principal values of the inverse trigonometric functions, the value of \[ \tan\left( \sin^{-1}\left(\frac{3}{5}\right) - 2 \cos^{-1}\left(\frac{2}{\sqrt{5}}\right) \right) \] is:

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

Hint:

When dealing with inverse trigonometric functions, convert the expressions to their respective trigonometric ratios for simpler computation.

Answer 1

The Correct Option is B

Let \[ 2 \cos^{-1}\left(\frac{2}{\sqrt{5}}\right) = \theta. \] Then, \[ \frac{2}{\sqrt{5}} = \cos\left(\frac{\theta}{2}\right), \quad \text{so} \quad \tan\left(\frac{\theta}{2}\right) = \frac{1}{2}. \] Now calculate: \[ \tan\left(\sin^{-1}\left(\frac{3}{5}\right)\right) = \frac{3}{4}, \quad \tan\left(\cos^{-1}\left(\frac{2}{\sqrt{5}}\right)\right) = \frac{4}{3}. \] Thus, \[ \tan\left(\sin^{-1}\left(\frac{3}{5}\right) - \cos^{-1}\left(\frac{2}{\sqrt{5}}\right)\right) = \frac{\frac{3}{4} - \frac{4}{3}}{1 + \frac{3}{4} \cdot \frac{4}{3}} = -\frac{7}{24}. \]