If $\tan \theta +\sec \theta =a$, then what is $\sec \theta$?

We know the standard trigonometric identity:
$${\sec }^2\theta -{\tan }^2\theta =1$$
This can be factored as:
$$(\sec \theta -\tan \theta )(\sec \theta +\tan \theta )=1$$
Given that $\tan \theta +\sec \theta =a$ (or $\sec \theta +\tan \theta =a$), we substitute this into the equation:
$$(\sec \theta -\tan \theta )\cdot a=1$$
$$\sec \theta -\tan \theta =\frac{1}{a}$$
Now, we have two simultaneous linear equations:$\sec \theta +\tan \theta =a$$\sec \theta -\tan \theta =\frac{1}{a}$ Adding equation (1) and equation (2):
$$2\sec \theta =a+\frac{1}{a}$$
$$2\sec \theta =\frac{a^2+1}{a}$$
$$\sec \theta =\frac{a^2+1}{2a}$$