Evaluate each of the following

Solution:

We have to find the following expression

$\tan ^{2} 30^{\circ}+\tan ^{2} 60^{\circ}+\tan ^{2} 45 \ldots \ldots(1)$

Now,

$\tan 30^{\circ}=\frac{1}{\sqrt{3}}, \tan 60^{\circ}=\sqrt{3}, \tan 45^{\circ}=1$

So by substituting above values in equation (1)

We get,

$\tan ^{2} 30^{\circ}+\tan ^{2} 60^{\circ}+\tan ^{2} 45$

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

$=\frac{1^{2}}{(\sqrt{3})^{2}}+(\sqrt{3})^{2}+1$

$=\frac{1}{3}+3+1$

$=\frac{1}{3}+4$

Now by taking LCM

We get,

$\tan ^{2} 30^{\circ}+\tan ^{2} 60^{\circ}+\tan ^{2} 45$

$=\frac{1}{3}+\frac{4 \times 3}{1 \times 3}$

$=\frac{1}{3}+\frac{12}{3}$

$=\frac{1+12}{3}$

$=\frac{13}{3}$

Therefore,

$\tan ^{2} 30^{\circ}+\tan ^{2} 60^{\circ}+\tan ^{2} 45^{\circ}=\frac{13}{3}$