Question

how to find the exact value of sec(13pi/6)

Answer 1

$\dfrac{13\pi}6\equiv\dfrac\pi6\mod2\pi$ (in other words, they represent the same angle on the unit circle)

So,

$\sec\dfrac{13\pi}6=\sec\dfrac\pi6=\dfrac1{\cos\dfrac\pi6}$

You have $\cos\dfrac\pi6=\dfrac{\sqrt3}2$, so $\sec\dfrac\pi6=\dfrac{\sqrt3}2$