Consider a simple pendulum with a bob of mass m that has an...

Let the length of the pendulum be $L$.

At the extreme position, the angle made by the string with the vertical is $\theta_0$.

The velocity of the bob at the extreme position is zero, so the centripetal acceleration is $v^2/L = 0$.

The forces acting on the bob along the direction of the string are the tension $T$ towards the support and the component of weight $mg \cos\theta_0$ away from the support.

Applying Newton's second law in the radial direction:

$T - mg \cos\theta_0 = \dfrac{mv^2}{L}$

Since $v = 0$ at the extreme position:

$T - mg \cos\theta_0 = 0$

$T = mg \cos\theta_0$