$$Cot^{-1}[\frac{\sqrt{1-sina}+\sqrt{1+sina}}{\sqrt{1-sina}-\sqrt{1+sina}}]=$$

$$Cot^{-1}[\frac{\sqrt{sin^2(a/2)+cos^2(a/2)-2sin(a/2)cos(a/2)}+\sqrt{sin^2(a/2)+cos^2(a/2)+2sin(a/2)cos(a/2)}}{\sqrt{sin^2(a/2)+cos^2(a/2)-2sin(a/2)cos(a/2)}-\sqrt{sin^2(a/2)+cos^2(a/2)+2sin(a/2)cos(a/2)}}]$$

$$Cot^{-1}[\frac{cos(a/2)-sin(a/2) + cos(a/2) + sin(a/2)}{cos(a/2)-sin(a/2) - cos(a/2) - sin(a/2)}]$$

$$Cot^{-1}[\frac{-2cos(a/2)}{2sin(a/2)}]$$

$$Cot^{-1}[Cot(-a/2)]$$

We know that $$Cot^{-1}[-x] = \pi - Cot^{-1}[x]$$

Therefore, $$Cot^{-1}[Cot(-a/2)]$$ = $$\pi - Cot^{-1}[Cot(a/2)]$$ = $$\pi-\frac{1}{2}a$$