Evaluate the half angle for tangent

The video tutorial explains how to calculate the tangent of 5π/8 using trigonometric identities. It discusses different formula options and emphasizes the importance of keeping one trigonometric function in the denominator for simplicity. The tutorial walks through the process of determining the value of U, calculating cosine and sine in the third quadrant, and simplifying the expression by rationalizing the denominator. The instructor highlights common mistakes and stresses the importance of checking algebraic steps.

How does the speaker suggest to handle the radicals in the denominator?

What is the final expression derived for the tangent function in the text?