Question

The value of \(2\tan^2 45^\circ + \cos^2 30^\circ - \sin^2 60^\circ\) is \dots

• A. 0
• B. 1
• C. 2
• D. 3

Hint:

Remember: \[ \sin 60^\circ = \cos 30^\circ \] So their squares are also equal. Whenever one is added and the other is subtracted, they cancel immediately.

Answer 1

The Correct Option is C

Concept: To simplify trigonometric expressions, we use standard trigonometric values. Important standard values: \[ \tan 45^\circ = 1 \] \[ \cos 30^\circ = \frac{\sqrt{3}}{2} \] \[ \sin 60^\circ = \frac{\sqrt{3}}{2} \] Also: \[ \cos^2 \theta = (\cos \theta)^2 \] \[ \sin^2 \theta = (\sin \theta)^2 \]

Step 1: Write the given expression. \[ 2\tan^2 45^\circ + \cos^2 30^\circ - \sin^2 60^\circ \]

Step 2: Substitute the standard values. Since: \[ \tan 45^\circ = 1 \] \[ \cos 30^\circ = \frac{\sqrt{3}}{2} \] \[ \sin 60^\circ = \frac{\sqrt{3}}{2} \] therefore: \[ 2(1)^2 + \left(\frac{\sqrt{3}}{2}\right)^2 - \left(\frac{\sqrt{3}}{2}\right)^2 \]

Step 3: Simplify the squares. \[ (1)^2 = 1 \] and: \[ \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} \] Thus: \[ 2(1) + \frac{3}{4} - \frac{3}{4} \]

Step 4: Simplify the expression further. Notice: \[ \frac{3}{4} - \frac{3}{4} = 0 \] So: \[ 2 + 0 \] \[ = 2 \]

Step 5: Final conclusion. Therefore: \[ 2\tan^2 45^\circ + \cos^2 30^\circ - \sin^2 60^\circ = 2 \] Final Answer: \[ \boxed{2} \]