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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,
$\tan 30^{\circ}=\frac{1}{\sqrt3}$
$\tan 45^{\circ}=1$
$\tan 60^{\circ}=\sqrt3$
Therefore,$ \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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