Question

use sum, difference, product, or half-angle formulas to find the exact value of each expression

$\tan 75°$

Short answer

$\tan 75°=2+\sqrt{3}$

Step-by-step solution

Step 1. Given expression

$\tan 75°$

Step 2. Using half angle formula 

We know :

$\tan \left(\frac{\alpha }{2}\right)=\pm \sqrt{\frac{1-\cos \alpha }{1+\cos \alpha }}$

Using it:
$$\begin{gathered} \tan \left(\frac{150}{2}\right)=\pm \sqrt{\frac{1-\cos 150°}{1+\cos 150°}} \\ \tan \left(75\right)=\sqrt{\frac{1-{\displaystyle \frac{-\sqrt{3}}{2}}}{1+{\displaystyle \frac{-\sqrt{3}}{2}}}} \\ \tan \left(75\right)=\sqrt{\frac{2+\sqrt{3}}{2-\sqrt{3}}} \end{gathered}$$

Step 3.Rationalizing

$$\begin{gathered} \tan (75°)=\left(\sqrt{\frac{2+\sqrt{3}}{2-\sqrt{3}}}\right).\left(\sqrt{\frac{2+\sqrt{3}}{2+\sqrt{3}}}\right) \\ \tan (75°)=\frac{2+\sqrt{3}}{\sqrt{2^2-{\left(\sqrt{3}\right)}^2}} \\ \tan (75°)=2+\sqrt{3} \end{gathered}$$