Question

If \( \sin^{-1}x + \sin^{-1}y = \frac{\pi}{6} \) and \( \cot^{-1}\left(\frac{1}{2}\right) - \cot^{-1}\left(\frac{1}{y}\right) = 0 \), then calculate \( 2x^2 + y^2 - xy = ? \) 

• A. \( \frac{1}{4} \)
• B. 1
• C. \( \frac{1}{2} \)
• D. 0

Hint:

For problems involving inverse trigonometric functions, use the standard identities and simplifications to solve for the unknowns.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We have a system of two equations involving inverse trigonometric functions. We need to solve for \( x \) and \( y \) and evaluate the expression.
Step 2: Key Formula or Approach:
1. Note that \( \cot^{-1}(1/x) = \tan^{-1} x \) for \( x>0 \).
2. From the second equation, find the relationship between \( x \) and \( y \).
3. Substitute into the first equation to find the values.
Step 3: Detailed Explanation:
1. From \( \cot^{-1}(1/x) - \cot^{-1}(1/y) = 0 \):
\( \cot^{-1}(1/x) = \cot^{-1}(1/y) \Rightarrow \frac{1}{x} = \frac{1}{y} \Rightarrow x = y \).
2. Substitute \( x = y \) into \( \sin^{-1} x + \sin^{-1} y = \pi/3 \):
\( 2\sin^{-1} x = \pi/3 \Rightarrow \sin^{-1} x = \pi/6 \).
\( x = \sin(\pi/6) = 1/2 \).
So, \( x = 1/2, y = 1/2 \).
3. Evaluate \( 2x^2 + y^2 - xy \):
\( = 2(1/2)^2 + (1/2)^2 - (1/2)(1/2) \)
\( = 2(1/4) + 1/4 - 1/4 \)
\( = 2/4 = 1/2 \).
Step 4: Final Answer:
The value of the expression is \( 1/2 \).