Question

The value of \( 2 \tan^2 45^\circ + \cos^2 30^\circ - \sin^2 60^\circ \) will be:

• A. 1
• B. \( \sqrt{2} \)
• C. 2
• D. None of these

Hint:

Always use the fundamental trigonometric values for standard angles like \( 30^\circ, 45^\circ, 60^\circ \) to simplify expressions.

Answer 1

The Correct Option is C

We know the following trigonometric values: - \( \tan 45^\circ = 1 \), so \( \tan^2 45^\circ = 1 \) - \( \cos 30^\circ = \frac{\sqrt{3}}{2} \), so \( \cos^2 30^\circ = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} \) - \( \sin 60^\circ = \frac{\sqrt{3}}{2} \), so \( \sin^2 60^\circ = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} \) Now substitute these values into the expression: \[ 2 \tan^2 45^\circ + \cos^2 30^\circ - \sin^2 60^\circ = 2 \times 1 + \frac{3}{4} - \frac{3}{4} \] Simplifying the expression: \[ = 2 + 0 = 2 \]
Step 1: Conclusion.
Thus, the value of the expression is 2.