Determine an exact value for the expression

Question

sec4π/3*cot5π/6-tan3π/4

Answer 1

4π/3 is in quad III (240°
so sec 4π/3 = -sec π/3
cos π/3 = 1/2, so sec 4π/3 = -2

5π/6 is in quad II (150°)
so cot 5π/6 = -cot π/6
we know tan π/3 = 1/√3
so cot 5π/6 = -√3

3π/4 is in quad II (135°)
tan 3π/4 = -1

sec4π/3*cot5π/6-tan3π/4
= sec(240°) cot(150°) - tan(135°)
= -2(-√3) - (-1)
= 2√3 + 1