How do you evaluate $\sec 780{}^\circ ?$

Hint: We use the reciprocal relation between each pair of trigonometric functions to find the value of the given trigonometric function. That is, $\sec 780{}^\circ :\,\,\sec x=\dfrac{1}{\cos x}.$ We will also use the result $\cos \left( n\pi +\theta \right)=\left\{ \begin{align}
  & \cos \theta ,\,\,\,\,\,\,\,\,if\,n\,is\,even \\
 & -\cos \theta ,\,\,\,if\,n\,is\,odd \\
\end{align} \right..$ This result is applicable if $180$ is put instead of $\pi ,$ since they are equivalent.

Complete step by step solution:
Let us consider the trigonometric function given here, $\sec 780{}^\circ .$
We are going to put $a=\sec 780{}^\circ .$
Calculation on the secant function can easily be done using the cosine function. Because, they are reciprocal of each other.
Let us recall this reciprocal relation of the secant function and the cosine function,
$\Rightarrow \sec x=\dfrac{1}{\cos x}.$
Now, with the help of the reciprocal relation, we will write,
$\Rightarrow a=\dfrac{1}{\cos 780{}^\circ }.$
And this can be rewritten as,
$\Rightarrow \cos 780{}^\circ =\dfrac{1}{a}$
Also, we will be able to write this as,
$\Rightarrow \cos 780{}^\circ =\dfrac{1}{\sec 780{}^\circ }.$
Now we consider $\cos 780{}^\circ .$
We know that $780=360+420.$
So, we will get the following,
$\Rightarrow \cos 780{}^\circ =\cos \left( 360+420 \right).$
If we apply the result $\cos \left( 360+\theta \right)=\cos \theta .$
Therefore, we get,
$\Rightarrow \cos 780{}^\circ =\cos 420{}^\circ .$
Furthermore, $420=360+60.$
Now, we will get the following,
$\Rightarrow \cos 780{}^\circ =\cos \left( 360+60 \right){}^\circ .$
We are applying the result $\cos \left( 360+\theta \right)=\cos \theta $ again. We get,
$\Rightarrow \cos 780{}^\circ =\cos 60{}^\circ .$
Now we know that $\cos 60{}^\circ =\dfrac{1}{2}.$
Thus, we get,
$\Rightarrow \cos 780{}^\circ =\dfrac{1}{2}.$
Remember what we have written earlier, $\cos 780{}^\circ =\dfrac{1}{a}.$
We surely understand that $\dfrac{1}{a}=\dfrac{1}{2}.$
Take the reciprocal of the whole equation, we get
$\Rightarrow a=2.$
 We have already put $a=\sec 780{}^\circ .$
We get the value of the given trigonometric identity,
$\Rightarrow \sec 780{}^\circ =2.$
Hence, the value of $\sec 780{}^\circ =2.$

Note: We can apply the result $\cos \left( 180n+\theta \right)=\left\{ \begin{align}
  & \cos \theta ,\,\,\,\,\,\,if\,n\,is\,even \\
 & -\cos \theta ,\,\,if\,n\,is\,odd \\
\end{align} \right.$ and directly get the answer as follows:
We consider $780=720+60.$
So, we will get,
$\Rightarrow \cos 780{}^\circ =\cos \left( 720+60 \right){}^\circ .$
Also, we know that $720$ is a multiple of $360.$
In this case $n$ is even, $720=180\times 4.$
So, we will get,
$\Rightarrow \cos 780{}^\circ =\cos 60{}^\circ .$
From this we can see that $\cos 60{}^\circ =\dfrac{1}{2}.$
Also, we will get,
$\Rightarrow \cos 780{}^\circ =\dfrac{1}{2}.$
And thus,
$\Rightarrow \cos 780{}^\circ =\dfrac{1}{\sec 780{}^\circ }=\dfrac{1}{2}.$
And we yield, $\sec 780{}^\circ =2.$