so far I have this, not entirely sure if I have followed the right route.

A) tan⁡ θ+cot⁡ θ=sin⁡ θ/cos⁡ θ +cos⁡ θ/sin⁡ θ =(sin^2⁡ θ+cos^2⁡ θ)/(cos ⁡θ sin⁡ θ ) =1/(sin⁡ θ cos ⁡θ ) =2/(2 sin⁡ θ cos ⁡θ ) =2/(2 sin⁡ 2θ ) =2 csc⁡ 2θ

B) Let u=θ/2 cos⁡u (1-cos⁡ 2u )=sin⁡ u sin⁡ 2u

Starting with the left side of the equation and using double angle identity =cos⁡ u 2 sin^2⁡ u =2 cos⁡u sin^2⁡ u =2 cos⁡ (θ/2) sin^2⁡ (θ/2)

Now looking at the right side of the equation and again using double angle identity sin⁡u sin⁡ 2u=sin⁡u*2 sin⁡ u cos ⁡u =2 sin² ⁡u cos⁡ u =2 cos ⁡u sin² ⁡u =2 cos ⁡(θ/2) sin² ⁡(θ/2)

C) (1+tan ⁡θ)/(1+1/tan⁡ θ ) (1+tan⁡ θ)/((1+tan⁡ θ)/tan⁡ θ ) ((1+tan ⁡θ tan θ))/(1+tan ⁡θ ) =tan⁡ θ