Evaluate each of the following: 4(sin^4 60° + cos64 30°) -3(tan^2 60° - tan^2 45°) + 5 cos^2 45°

Question

Evaluate each of the following:

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

Answer 1

Given: 4(sin⁴ 60° + cos⁴ 30°) -3(tan² 60° - tan² 45°) + 5 cos² 45°

To find: The value of 4 (sin⁴ 60° + cos⁴ 30°) - 3 (tan² 60° - tan² 45°) + 5cos² 45°. 

Solution: 

We know,

sin 60° = \(\frac{\sqrt3}2\), cos 30° = \(\frac{\sqrt3}2\), cos 45° = \(\frac{1}{\sqrt2}\), tan 45° = 1, tan 60° = \(\sqrt3\)

Substitute the above values in 4 (sin⁴ 60° + cos⁴ 30°) - 3 (tan² 60° - tan² 45°) + 5cos² 45°, 

Solve,

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