SOLUTION :-
Given that,
θ € 4th quadrant
3π/2 < θ < 2π
And
sin θ = - 1/2
We know,
sin²θ + cos²θ = 1
[ -1/2] ² + cos²θ = 1
1/4 + cos²θ = 1
cos²θ = 1 - 1/4
cos²θ = 4 - 1 / 4
cos²θ = 3/4
cos²θ = + √3/2
As,
θ € 4th quadrant = cos θ > 0
cosθ = √3/2
tan θ = sin θ / cos θ
tan θ = 1/2 ÷ √3/2
tan θ = - 1/√3
Now,
cosec θ = 1/sin θ = -2
sec θ = 1/cos θ = 2√3
cot θ = 1/tan θ = - √3