Certainly! Let's go through the problem step-by-step.
We are given the equation:
[tex]\[ \sec \theta + \cos \theta = \frac{5}{2} \][/tex]
Our goal is to find the value of \( \sec \theta - \cos \theta \).
First, let's isolate \( \sec \theta \) from the given equation:
[tex]\[ \sec \theta = \frac{5}{2} - \cos \theta \][/tex]
Now we will use this expression for \( \sec \theta \) to find \( \sec \theta - \cos \theta \):
[tex]\[ \sec \theta - \cos \theta = \left( \frac{5}{2} - \cos \theta \right) - \cos \theta \][/tex]
Simplify the expression on the right-hand side:
[tex]\[ \sec \theta - \cos \theta = \frac{5}{2} - \cos \theta - \cos \theta \][/tex]
[tex]\[ \sec \theta - \cos \theta = \frac{5}{2} - 2 \cos \theta \][/tex]
Therefore, the value of \( \sec \theta - \cos \theta \) is:
[tex]\[ \sec \theta - \cos \theta = \frac{5}{2} - 2 \cos \theta \][/tex]
Hence, using the relation derived from the given condition, the expression simplifies to:
[tex]\[ \boxed{\frac{5}{2} - 2 \cos \theta} \][/tex]
