The product (1 + cotθ − cosecθ)(1 + tanθ + secθ) simplifies to:

Write everything over sinθ and cosθ: the first bracket is (sinθ + cosθ − 1)/sinθ and the second is (cosθ + sinθ + 1)/cosθ. The numerators multiply to $(sin\theta +cos\theta {)}^2-1=1+2$sinθ·cosθ − 1 = 2sinθ·cosθ, which divided by sinθ·cosθ leaves 2. Check at θ = 30°: $(1+1.732-2)(1+0.577+1.155)=0.732\times 2.732=2$.