Question

Evaluate: `(3 cos^2 30^circ - 6  "cosec"^2 30^circ)/(tan^2 60^circ)`

Answer 1

Substitute the standard values for trigonometric ratios of 30° and 60° into the expression.

`cos 30^circ = sqrt(3)/2`

`"cosec"  30^circ = 1/(sin 30^circ) = 1/(1/2) = 2`

`tan 60^circ = sqrt(3)`

Substitute the values into the expression:

Expression = `(3(sqrt(3)/2)^2 - 6(2)^2)/(sqrt(3))^2`

Simplify the numerator and denominator:

= `(3(3/4) - 6(4))/3`

= `(9/4 - 24)/3`

Find a common denominator for the numerator:

= `((9 - 96)/4)/3`

= `((-87)/4)/3`

Divide by 3:

= `- 87/(4 xx 3)`

= ` - 87/12`

= `- 29/4`

The evaluated value is `- 29/4`.