Question

Evaluate : \(\frac{3 \cos^2 30^{\circ} - 6 \csc^2 30^{\circ}}{\tan^2 60^{\circ}}\).

Hint:

Double check the square terms! It's common to forget to square the numbers in the denominator of the fraction (like squaring \(2\) to get \(4\) in \(\cos^2 30^{\circ}\)).

Answer 1

Step 1: Write Standard Trigonometric Values
cos 30° = √3 / 2
csc 30° = 2
tan 60° = √3

Step 2: Substitute into the Expression
Expression = [3(cos²30°) − 6(csc²30°)] / (tan²60°)

= [3( (√3/2)² ) − 6(2²)] / ( (√3)² )

Step 3: Simplify Step by Step
(√3/2)² = 3/4
2² = 4
(√3)² = 3

So expression becomes:
= [3(3/4) − 6(4)] / 3
= [9/4 − 24] / 3

Convert 24 into fraction:
24 = 96/4

So numerator:
= (9/4 − 96/4)
= −87/4

Now divide by 3:
= (−87/4) ÷ 3
= −87 / 12
= −29 / 4

Final Answer:
The value of the expression is −29/4