Questions: 2 tan (π/8) / (1 - tan^2 (π/8))

Transcript text: $\frac{2 \tan \frac{\pi}{8}}{1-\tan ^{2} \frac{\pi}{8}}$

Solution

Solution Steps

To solve the expression $\frac{2 \tan \frac{\pi}{8}}{1-\tan ^{2} \frac{\pi}{8}}$, we can use the double angle identity for tangent. The identity states that $\tan(2\theta) = \frac{2\tan(\theta)}{1-\tan^2(\theta)}$. By recognizing that the given expression matches this identity, we can simplify it to $\tan\left(2 \times \frac{\pi}{8}\right)$, which is $\tan\left(\frac{\pi}{4}\right)$.

Step 1: Identify the Expression

We start with the expression

\[ \frac{2 \tan \frac{\pi}{8}}{1 - \tan^2 \frac{\pi}{8}}. \]

Step 2: Apply the Double Angle Identity

Using the double angle identity for tangent, we know that

\[ \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}. \]

In our case, let $\theta = \frac{\pi}{8}$. Thus, we can rewrite the expression as

\[ \tan\left(2 \times \frac{\pi}{8}\right) = \tan\left(\frac{\pi}{4}\right). \]

Step 3: Evaluate the Tangent

We know that

\[ \tan\left(\frac{\pi}{4}\right) = 1. \]