Pre-calc question

Question

Evaluate cos 5𝜋/12 + sin𝜋/12 / cos𝜋/12 - sin 𝜋/12

(its two factions over two fractions )

Answer 1

cos(5π/12) = sin(π/12)

cos(π/12) - sin(π/12) = √2[(1/√2)cos(π/12) - (1/√2)sin(π/12) ] = √2(sin(π/4)cos(-π/12) + cos(π/4)sin(-π/12)) =

√2 sin(π/4-π/12) = √2 sin(π/6)

(cos(5π/12) + sin(π/12))/(cos(π/12) - sin(π/12)) =

2sin(π/12)/√2 sin(π/6) =

2sin(π/12)/(2*√2 sin(π/12)cos(π/12)) =

1/(√2cos(π/12)) =

1/√(1+cos((π/6)) =

1/√(1+√3/2) =

2√(1-√3/2)