Question

Find the value of:

`(2 cos 30°)/(cos^2 45°) + (tan 60°)/(sin 60° xx tan 30°)`

Answer 1

Given:

`(2 cos 30°)/(cos^2 45°) + (tan 60°)/(sin 60° xx tan 30°)`

Step 1: Recall standard values

cos 30° = `sqrt3/2`

cos 45° = `1/sqrt2 → cos^2 45° = 1/2`

tan 60° = `sqrt3`

sin 60° = `sqrt3/2`

tan 30° = `1/sqrt3`

Step 2:

`(2. sqrt3/2)/(1/2) = sqrt3/(1/2) = sqrt3 xx 2 = 2sqrt3`

Step 3:

sin 60° × tan 30° = `sqrt3/2 xx 1/sqrt3 = 1/2`

`(tan 60°)/(sin 60° xx tan 30°) = sqrt3/(1/2) = sqrt3 xx 2 = 2sqrt3` 

= `2 sqrt3 + 2sqrt3`  ...[Adding both results]

= `4sqrt3`