Question

Evaluate:

`(2 tan 30^circ sec 60^circ tan 45^circ)/(1 - sin^2 60^circ)`

Answer 1

Given: `(2 tan 30^circ sec 60^circ tan 45^circ)/(1 - sin^2 60^circ)`

Step-wise calculation:

1. Basic values:

`tan 30^circ = 1/sqrt(3)`

`sec 60^circ = 1/(cos 60^circ) = 1/(1/2) = 2`

tan 45° = 1

`sin 60^circ = sqrt(3)/2`

⇒ `sin^2 60^circ = 3/4`

2. Numerator:

`2 xx (1/sqrt(3)) xx 2 xx 1`

= `4/sqrt(3)`

3. Denominator:

`1 - sin^2 60^circ = 1 - 3/4`

= `1/4`

4. Division: `(4/sqrt(3)) ÷ (1/4)`

= `(4/sqrt(3)) xx 4`

= `16/sqrt(3)` Rationalized form: `(16sqrt(3))/3`