Evaluate (tan(pi/3))/2-1/(sec(pi/6))

$\frac{\tan (\frac{π}{3})}{2} - \frac{1}{\sec (\frac{π}{6})}$

Simplify each term.

The exact value of $\tan (\frac{π}{3})$ is $\sqrt{3}$ .

$\frac{\sqrt{3}}{2} - \frac{1}{\sec (\frac{π}{6})}$

Rewrite $\sec (\frac{π}{6})$ in terms of sines and cosines.

$\frac{\sqrt{3}}{2} - \frac{1}{\frac{1}{\cos (\frac{π}{6})}}$

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (\frac{π}{6})}$ .

$\frac{\sqrt{3}}{2} - (1 \cos (\frac{π}{6}))$

Multiply $\cos (\frac{π}{6})$ by $1$ .

$\frac{\sqrt{3}}{2} - \cos (\frac{π}{6})$

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$\frac{\sqrt{3}}{2} - \frac{\sqrt{3}}{2}$

Simplify terms.

Combine the numerators over the common denominator.

$\frac{\sqrt{3} - \sqrt{3}}{2}$

Subtract $\sqrt{3}$ from $\sqrt{3}$ .

$\frac{0}{2}$

Divide $0$ by $2$ .

$0$