Evaluate: $\dfrac{2\tan^2 45^\circ + \cos^2 30^\circ - \sin^2 60^\circ}{ \tan^2 60^\circ}$.
Step-by-Step Solution
Key Concept: Substitute all standard values carefully and simplify step by step.
$\tan45^\circ=1,\ \cos30^\circ=\dfrac{\sqrt3}{2},\ \sin60^\circ=\dfrac{\sqrt3}{2},\ \tan60^\circ=\sqrt3$. [1.0 Mark]
Numerator $=2(1)^2+\left(\dfrac{\sqrt3}{2}\right)^2-\left(\dfrac{\sqrt3}{2}\right)^2=2+0=2$ (since the last two terms are equal and cancel). [1.0 Mark]
Denominator $=(\sqrt3)^2=3$. So the expression $=\dfrac23$. [1.0 Mark]
Correct Answer: