Question

use an appropriate half - angle formula to find the exact value of the expression. tan(15°)

Explanation:

Step1: Recall half - angle formula for tangent

The half - angle formula for tangent is $\tan\frac{\alpha}{2}=\frac{1 - \cos\alpha}{\sin\alpha}$. We know that $15^{\circ}=\frac{30^{\circ}}{2}$, so $\alpha = 30^{\circ}$.

Step2: Substitute values of $\sin\alpha$ and $\cos\alpha$

We know that $\sin30^{\circ}=\frac{1}{2}$ and $\cos30^{\circ}=\frac{\sqrt{3}}{2}$. Substituting into the formula $\tan15^{\circ}=\tan\frac{30^{\circ}}{2}=\frac{1-\cos30^{\circ}}{\sin30^{\circ}}=\frac{1 - \frac{\sqrt{3}}{2}}{\frac{1}{2}}$.

Step3: Simplify the expression

$\frac{1 - \frac{\sqrt{3}}{2}}{\frac{1}{2}}=\frac{\frac{2-\sqrt{3}}{2}}{\frac{1}{2}}=2 - \sqrt{3}$.

Answer:

$2-\sqrt{3}$