Question

The value of $\tan 75°$ is

• A. $2+\sqrt{3}$
• B. $2-\sqrt{3}$
• C. $1+\sqrt{3}$
• D. $\sqrt{3}-2$

Answer 1

The correct option is A

$2+\sqrt{3}$

First we write the given angle in the sum of two standard angles such that $75=45+30$

So $\tan 75°=\tan (45°+30°)$

Now we use the formula $\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}$

So we have $$\begin{gathered} \tan (45°+30°)=\frac{\tan 45°+\tan 30°}{1-\tan 45°\tan 30°}(\tan 45°=1,\tan 30°=\frac{1}{\sqrt{3}}) \\ =\frac{1+{\displaystyle \frac{1}{\sqrt{3}}}}{1-1.{\displaystyle \frac{1}{\sqrt{3}}}}=\frac{\sqrt{3}+1}{\sqrt{3}}\times \frac{\sqrt{3}}{\sqrt{3}-1}=\frac{\sqrt{3}+1}{\sqrt{3}-1} \end{gathered}$$

Now we rationalise the denominator and get

$\frac{\sqrt{3}+1}{\sqrt{3}-1}\times \frac{\sqrt{3}+1}{\sqrt{3}+1}=\frac{{(\sqrt{3}+1)}^2}{{\left(\sqrt{3}\right)}^2-1^2}=\frac{3+1+2\sqrt{3}}{3-1}=\frac{4+2\sqrt{3}}{2}=\frac{2(2+\sqrt{3})}{2}=2+\sqrt{3}$

Hence option(A) is the correct answer and all other options are incorrect answers.