[ (4/3) tan2 60° + 3 cos2 30° − 2 sec2 30° − (3/4) cot2 60° ] ÷ [ sin 60° × cos 30° − cos 60° × sin 30° ]Find the value of:
6 2⁄3
Step 1: Simplify the terms
We use trigonometric identities:
tan 60° = √3, cot 60° = 1/√3, sec 30° = 2/√3, sin 60° = √3/2, cos 30° = √3/2, and cos 60° = 1/2.
Substitute these values:
Numerator = (4/3)(√3)2 + 3(√3/2)2 − 2(2/√3)2 − (3/4)(1/√3)2
= (4/3)(3) + 3(3/4) − 2(4/3) − (3/4)(1/3)
= 4 + 9/4 − 8/3 − 1/4
Denominator = sin 60° cos 30° − cos 60° sin 30°
= (√3/2)(√3/2) − (1/2)(1/2)
= 3/4 − 1/4 = 2/4 = 1/2.
Step 2: Divide numerator by denominator
Simplify to get 6 2⁄3.
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