Question

If \(x = \cos 30^\circ - \sin 30^\circ\) and \(y = \tan 60^\circ - \cot 60^\circ\), then

• A. \(x = y\)
• B. \(x>y\)
• C. \(x<y\)
• D. \(x>1, y<1\)

Hint:

Substitute exact trigonometric values for standard angles carefully to compare expressions.

Answer 1

The Correct Option is B

Given:
\[ x = \cos 30^\circ - \sin 30^\circ, \quad y = \tan 60^\circ - \cot 60^\circ \]

Step 1: Calculate \(x\)
\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \approx 0.866, \quad \sin 30^\circ = \frac{1}{2} = 0.5 \] \[ x = 0.866 - 0.5 = 0.366 \]

Step 2: Calculate \(y\)
\[ \tan 60^\circ = \sqrt{3} \approx 1.732, \quad \cot 60^\circ = \frac{1}{\sqrt{3}} \approx 0.577 \] \[ y = 1.732 - 0.577 = 1.155 \]

Step 3: Compare \(x\) and \(y\)
\[ x = 0.366, \quad y = 1.155 \] So, \[ x < y \]

Note: The correct answer provided is \(x > y\), but calculation shows \(x < y\).
Please verify the problem or values.

Final Result:
Based on calculations, \[ x < y \]