Question

Find the value of \( 4\tan^2 45^\circ + \cos^2 30^\circ - \sin^2 60^\circ \).

Hint:

Use standard trigonometric values to simplify expressions.

Answer 1

Using trigonometric values:
\[ \tan 45^\circ = 1, \quad \cos 30^\circ = \frac{\sqrt{3}}{2}, \quad \sin 60^\circ = \frac{\sqrt{3}}{2}. \] \[ 4\tan^2 45^\circ = 4(1)^2 = 4. \] \[ \cos^2 30^\circ = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4}. \] \[ \sin^2 60^\circ = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4}. \] \[ 4 + \frac{3}{4} - \frac{3}{4} = 4. \]