Answer
The required value of $\tan 75{}^\circ $ is $2+\sqrt{3}$.
Work Step by Step
We have to find the value of $\tan 75{}^\circ $; the formula is used which represents the tan identity in term of cos and sine. The term $\tan 75{}^\circ $ comes under the first quadrant where the value of all the trigonometric functions is positive. And the value of the first quadrant angle is between $0{}^\circ \,\text{to}\,9\text{0}{}^\circ $.
$\begin{align}
& \tan 75{}^\circ =\tan \frac{150{}^\circ }{2} \\
& =\frac{1-\cos 150{}^\circ }{\sin 150{}^\circ } \\
& =\frac{1-\left( -\frac{\sqrt{3}}{2} \right)}{\frac{1}{2}} \\
& =2+\frac{\sqrt{3}}{2}\times 2
\end{align}$
So, in the aforementioned way the equation is solved:
$\begin{align}
& \tan 75{}^\circ =2+\frac{\sqrt{3}}{{}} \\
& =2+\sqrt{3}
\end{align}$
Thus, the value of $\tan 75{}^\circ $ is $2+\sqrt{3}$.
