Question

Evaluate the following:

4(sin⁴ 60° + cos⁴ 30°) – 3(tan² 60° – tan² 45°) + 5 cos² 45°

Answer 1

4(sin⁴ 60° + cos⁴ 30°) − 3(tan² 60° − tan² 45°) + 5 cos² 45°  ........(i)

By trigonometric ratios we have

`sin 60^@ = sqrt3/2 - cos 30^@(sqrt3/2)   tan 60^@ = sqrt3   tan 45^@ = 1   cos 45^@ = 1/sqrt2`

By substituting above values in (i), we get

`4([sqrt3/2]^4) + [sqrt3/2]^4) - 3([3]^2 - [1]^2) + 5[1/sqrt2]^2` 

`=> 4[9/16 + 9/16] - 3[3 - 1]+ 5[1/2]`

`=> 4 · 18/16 - 6 + 5/2`

`=> 1/4 - 6 + 5/2`

`=> 1/4 - 6  + 5/2`

`= 9/2 + 5/2 - 6`

`= 14/2 - 6 = 7 - 6 = 1`