Question

how to prove that the area of a trapezoid is given by the following function : A= (2-sinθ)/(2cosθ)

Answer 1

since when do trapezoids involve θ?
I think you have omitted some vital information.

Answer 2

point T is located on the unit circle whose equation: x ^2 + y ^2 =1 , at the angle θ from the positive x axis where 0 ≤ θ<π /2 as in the adjacent figure:
1)prove that the area of a trapezoid OBQP is given by the following function : A= (2-sinθ)/(2cosθ)

2) Prove that the equation of the straight line PT is: x cosθ+y sin θ=1

3)Find the measure of the angle θ for which the area of the trapezoid is the least.