which of the following is equivalent to tan(5π/6)? tan(-5π/6) tan(-π/6) tan(7π/6) cot(5π/6)

Explanation:

Step1: Recall the periodicity of tangent function

The tangent function $y = \tan(x)$ has a period of $\pi$, i.e., $\tan(x)=\tan(x + k\pi)$ for any real - number $x$ and integer $k$.

Step2: Rewrite $\tan(\frac{5\pi}{6})$ using the period

We know that $\frac{5\pi}{6}=\pi-\frac{\pi}{6}$, and $\tan(x)=\tan(x + \pi)$. Also, $\tan(\frac{5\pi}{6})=\tan(\frac{5\pi}{6}-\pi)=\tan(-\frac{\pi}{6}+\pi)=\tan(-\frac{\pi}{6})$. We can also check other options:

• $\tan(-\frac{5\pi}{6})=-\tan(\frac{5\pi}{6})$ (since $\tan(-x)=-\tan(x)$).
• $\tan(\frac{7\pi}{6})=\tan(\pi+\frac{\pi}{6})=\tan(\frac{\pi}{6})$ (using the period property $\tan(x + \pi)=\tan(x)$).
• $\cot(\frac{5\pi}{6})=\frac{1}{\tan(\frac{5\pi}{6})}$.

Answer:

$\tan(-\frac{\pi}{6})$