Answer:
C
Step-by-step explanation:
If you notice, this is what the tangent-difference identity resembles. The tangent-difference identity is:
[tex]\tan(\alpha-\beta)=\frac{\tan(\alpha)-\tan(\beta)}{1-\tan(\alpha)\tan(\beta)}[/tex]
We have the expression:
[tex]\frac{\tan(\frac{\pi}{7})-\tan(\frac{\pi}{8})}{1-\tan(\frac{\pi}{7})\tan(\frac{\pi}{8})}[/tex]
So, our α is π/7 and our β is π/8. Therefore:
[tex]\frac{\tan(\frac{\pi}{7})-\tan(\frac{\pi}{8})}{1-\tan(\frac{\pi}{7})\tan(\frac{\pi}{8})}=\tan(\frac{\pi}{7}-\frac{\pi}{8}})[/tex]
Simplify:
[tex]\frac{\tan(\frac{\pi}{7})-\tan(\frac{\pi}{8})}{1-\tan(\frac{\pi}{7})\tan(\frac{\pi}{8})}=\tan(\frac{8\pi}{56}-\frac{7\pi}{56}})[/tex]
Subtract:
[tex]\frac{\tan(\frac{\pi}{7})-\tan(\frac{\pi}{8})}{1-\tan(\frac{\pi}{7})\tan(\frac{\pi}{8})}=\tan(\frac{\pi}{56})[/tex]
So, our answer is C.
And we're done!
