1. **State the problem:** Evaluate $\sec \frac{\pi}{4}$ without using a calculator. 2. **Recall the definition:** The secant function is the reciprocal of the cosine function, so $$\sec x = \frac{1}{\cos x}$$ 3. **Evaluate cosine at $\frac{\pi}{4}$:** We know from the unit circle that $$\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}$$ 4. **Calculate secant:** Using the reciprocal, $$\sec \frac{\pi}{4} = \frac{1}{\cos \frac{\pi}{4}} = \frac{1}{\frac{\sqrt{2}}{2}} = \frac{2}{\sqrt{2}}$$ 5. **Simplify the expression:** Multiply numerator and denominator by $\sqrt{2}$ to rationalize the denominator, $$\frac{2}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{2\sqrt{2}}{2} = \sqrt{2}$$ 6. **Final answer:** $$\sec \frac{\pi}{4} = \sqrt{2}$$ Therefore, the correct choice is option 4: $\sqrt{2}$.