Question

what is work done by force of tension in string of a pendulum

Answer 1

The reason is not complicated and that's why many textbook simply skip the calculation. The work done is defined as

W=∫T⃗ ⋅ds⃗ W=∫T→⋅ds→

And not that the tension on a simple pendulum is always perpendicular to the displacement vector. Therefore, the dot product is 0 and thus the line integral is 0.