F.4 m2 --- trigonometry

a) Find the value of sinθ - cosθ
(sinθ + cosθ)^2 = (-1/2)^2
1 + 2sinθcosθ = 1/4
sin2θ = - 3/4
and
(sinθ - cosθ)^2
= 1 - sin2θ
= 1+3/4 = 7/4
sosinθ - cosθ = -√7 /2 or √7 /2 (rejected since sinθ is - ,cosθ is + for
(3π) / 2 < θ < 2π )
b)Hence, find the value of sec(π + θ)
sec(π + θ) = - 1 / cos θ
and
sinθ + cosθ = -1/2...(1)
sinθ - cosθ = -√7 /2...(2)
(1)-(2) :
2cosθ = -1/2 + √7 /2
cosθ = (√7 - 1) / 4
so
sec(π + θ) = - 1 / cos θ = - 1 / [(√7 - 1)/4]

= 4 / (1 - √7)

= 4(1 + √7) / - 6

= - 2(1 + √7)/3