Evaluate each of the following:$ \tan ^{2} 30^{\circ}+\tan ^{2} 60^{\circ}+\tan ^{2} 45^{\circ} $

Given:

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

To do:

We have to evaluate \( \tan ^{2} 30^{\circ}+\tan ^{2} 60^{\circ}+\tan ^{2} 45^{\circ} \).

Solution:  

We know that,

$\tan 30^{\circ}=\frac{1}{\sqrt3}$

$\tan 45^{\circ}=1$

$\tan 60^{\circ}=\sqrt3$

$ \tan ^{2} 30^{\circ}+\tan ^{2} 60^{\circ}+\tan ^{2} 45^{\circ}=\left(\frac{1}{\sqrt3}\right)^{2} +(\sqrt{3})^{2} +(1)^{2}$

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

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

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

$=\frac{13}{3}$

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

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Updated on: 10-Oct-2022

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