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: Conversion
Convert inverse functions to $\tan^{-1}$.
$\sin^{-1}(3/5) = \tan^{-1}(3/4)$.
$\cos^{-1}(2/\sqrt{5}) = \tan^{-1}(1/2)$.

Step 2: Analysis
The expression is $\tan(\tan^{-1}(3/4) - 2\tan^{-1}(1/2))$.
Calculate $2\tan^{-1}(1/2) = \tan^{-1}(\frac{2(1/2)}{1-(1/2)^2}) = \tan^{-1}(1/(3/4)) = \tan^{-1}(4/3)$.

Step 3: Calculation
$\tan(\tan^{-1}(3/4) - \tan^{-1}(4/3)) = \frac{3/4 - 4/3}{1 + (3/4)(4/3)}$
$= \frac{(9-16)/12}{1+1} = \frac{-7/12}{2} = -7/24$.

Step 4: Conclusion
Hence, the value is $-7/24$. Final Answer: (B)