Find the value of $\tan \frac{\pi}{8}$.

Step 1: Use the half-angle identity for tangent

To find the value of $\tan \frac{\pi}{8}$, we can use the half-angle identity for tangent. In this case, we can set $\frac{\theta}{2} = \frac{\pi}{8}$, which means $\theta = \frac{\pi}{4}$. This identity allows us to express the tangent of a half-angle in terms of trigonometric functions of the full angle.

Step 2: Substitute the value of $\theta$

Now, we substitute $\theta = \frac{\pi}{4}$ into the half-angle identity. This gives us an expression for $\tan \frac{\pi}{8}$ in terms of $\cos \frac{\pi}{4}$ and $\sin \frac{\pi}{4}$.

Step 3: Evaluate $\sin \frac{\pi}{4}$ and $\cos \frac{\pi}{4}$

We know the exact values for the sine and cosine of $\frac{\pi}{4}$ (or $45^\circ$). Both $\sin \frac{\pi}{4}$ and $\cos \frac{\pi}{4}$ are equal to $\frac{1}{\sqrt{2}}$.

Step 4: Substitute the values and simplify

Substitute the values of $\sin \frac{\pi}{4}$ and $\cos \frac{\pi}{4}$ into the expression. Then, simplify the complex fraction by multiplying the numerator and denominator by $\sqrt{2}$. This leads to the final simplified value for $\tan \frac{\pi}{8}$.

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