Question

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

• A. $7/24$
• B. $-7/24$
• C. $1/24$
• D. $-1/24$

Hint:

Always convert inverse trigonometric functions to $\tan^{-1}$ for easier addition/subtraction.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
Evaluate the tangent of a combination of inverse trigonometric functions by converting them to the tangent domain.

Step 2: Key Formula or Approach:
1. Conversion: $\sin^{-1}(3/5) = \tan^{-1}(3/4)$.
2. Conversion: $\cos^{-1}(2/\sqrt{5}) = \tan^{-1}(1/2)$.
3. Double angle: $\tan(2\theta) = \frac{2 \tan \theta}{1 - \tan^2 \theta}$.
4. Difference: $\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$.

Step 3: Detailed Explanation:
Let $A = \sin^{-1}(3/5) \Rightarrow \tan A = 3/4$.
Let $\theta = \cos^{-1}(2/\sqrt{5}) \Rightarrow \tan \theta = 1/2$.
Let $B = 2\theta = 2 \tan^{-1}(1/2)$.
\[ \tan B = \frac{2(1/2)}{1 - (1/2)^2} = \frac{1}{1 - 1/4} = \frac{1}{3/4} = \frac{4}{3} \] We need $\tan(A - B)$: \[ \tan(A - B) = \frac{3/4 - 4/3}{1 + (3/4)(4/3)} = \frac{(9 - 16)/12}{1 + 1} = \frac{-7/12}{2} = -\frac{7}{24} \]

Step 4: Final Answer:
The value is $-\frac{7}{24}$.