Question

Evaluate: `(5cos^2 60° +  4sec^2 30° - tan^2 45°)/(sin^2 30° + sin^2 60°)`

Answer 1

We will use the known values of trigonometric functions at specific angles:

cos 60° = `1/2`

sec 30° = `1/(cos 30°) = 2/sqrt3`

tan 45° = 1

Calculating the numerator:

`5cos^2 60° + 4sec^2 30° − tan^2 45°` = `5 (1/2​)^2 + 4(​2/sqrt3​)^2 − 1^2`

= `5(1/4) + 4(4/3) − 1`

= `5/4 + 16/3 − 1`

Calculating the denominator:

`sin^2 30° + sin^2 60° = (1/2)^2 + (sqrt3/2)^2`

`= 1/4 + 3/4`

= 1

Therefore, the expression simplifies to:

`((5/4) + 16/3 - 1)/1`

= `5/4 + 16/3 - 1`

Converting the fractions to a common denominator and simplifying:

= `(15 + 64 - 12)/12`

= `67/12`

So, the evaluated result is `67/12`.