Solving the Trigonometric Equation

We are given the equation:

$ \cos x + \frac{1}{\cos x} = 2 $

Let $y = \cos x$. Substituting this into the equation gives:

$ y + \frac{1}{y} = 2 $

To solve for $y$, we multiply both sides by $y$ (assuming $y \neq 0$):

$ y^2 + 1 = 2y $

Rearranging the terms, we get a quadratic equation:

$ y^2 - 2y + 1 = 0 $

This is a perfect square trinomial:

$ (y - 1)^2 = 0 $

Solving for $y$, we find:

$ y = 1 $

Since we let $y = \cos x$, this means:

$ \cos x = 1 $

Evaluating the Expression

Now we need to find the value of $\cos^n x + \frac{1}{\cos^n x}$.

We substitute the value $\cos x = 1$ into the expression:

• $\cos^n x = (1)^n = 1$ (for any positive integer $n$)
• $\frac{1}{\cos^n x} = \frac{1}{(1)^n} = \frac{1}{1} = 1$

Therefore, the value of the expression is:

$ \cos^n x + \frac{1}{\cos^n x} = 1 + 1 = 2 $