Question

The value of $(\sqrt{3}+\tan 1^{\circ })(\sqrt{3}+\tan 2^{\circ })(\sqrt{3}+\tan 3^{\circ })\cdots (\sqrt{3}+\tan {28}^{\circ })(\sqrt{3}+\tan {29}^{\circ })$ is

• A. $2^{28}$
• B. $2^{15}+2-\sqrt{3}$
• C. $2^{29}$
• D. $2^{14}+2-\sqrt{3}$

Answer 1

The correct option is C $2^{29}$

If $A+B={30}^{\circ }$
$\tan (A+B)=\frac{1}{\sqrt{3}}$
$\Rightarrow \frac{\tan A+\tan B}{1-\left(\tan A\right)\left(\tan B\right)}=\frac{1}{\sqrt{3}}$
$\Rightarrow (\sqrt{3}+\tan A)(\sqrt{3}+\tan B)=4$

$\therefore (\sqrt{3}+\tan 1^{\circ })(\sqrt{3}+\tan 2^{\circ })(\sqrt{3}+\tan 3^{\circ })\cdots (\sqrt{3}+\tan {28}^{\circ })(\sqrt{3}+\tan {29}^{\circ })$
$=4^{14}(\sqrt{3}+\tan {15}^{\circ })$
$=4^{14}(\sqrt{3}+(2-\sqrt{3}))=2^{29}$