Question

question 16 (5 points) evaluate the expression. $\frac{\tan\frac{pi}{7}-\tan\frac{pi}{8}}{1 + \tan\frac{pi}{7}cdot\tan\frac{pi}{8}}$ a) $\tan\frac{pi}{56}$ b) $-\tan\frac{pi}{56}$ c) $\tan\frac{17pi}{56}$ d) $-\tan\frac{17pi}{56}$ question 17 (5 points)

Explanation:

Step1: Recall the tangent difference - formula

The formula for $\tan(A - B)$ is $\frac{\tan A-\tan B}{1 + \tan A\tan B}$.

Step2: Identify $A$ and $B$

In the given expression $\frac{\tan\frac{\pi}{7}-\tan\frac{\pi}{8}}{1+\tan\frac{\pi}{7}\tan\frac{\pi}{8}}$, we have $A=\frac{\pi}{7}$ and $B = \frac{\pi}{8}$.

Step3: Calculate $A - B$

$A - B=\frac{\pi}{7}-\frac{\pi}{8}=\frac{8\pi-7\pi}{56}=\frac{\pi}{56}$.

Step4: Apply the formula

Since $\frac{\tan A-\tan B}{1 + \tan A\tan B}=\tan(A - B)$, substituting $A=\frac{\pi}{7}$ and $B=\frac{\pi}{8}$ gives $\tan(\frac{\pi}{7}-\frac{\pi}{8})=\tan\frac{\pi}{56}$.

Answer:

A. $\tan\frac{\pi}{56}$