why tanx/2.cotx/2 is not equal to 1 when x= 0 and if equal then see my attached image

first of all your ques is wrong. we have to find tanx/2.coty/2 and not tanx/2.cotx/2. you can apply componendo and dividendo, and then use the formula (1 – cosz)/(1+cosz)= tan^2(z/2). thence you will obtain the answer root3.

when x=0, obviously x/2 is also zero.

so tan(x/2) is also zero.

but cotx/2 is not defined at x/2=0

it is because cot(theta)= cos(theta)/sin(theta), and when theta is zero, sintheta is also zero and division by zero is not allowed (or defined).