Question
Without using tables, evaluate the following: $\tan ^2 30^{\circ}+\tan ^2 60^{\circ}+\tan ^2 45^{\circ}$
✓
Answer
$ \tan ^2 30^{\circ}+\tan ^2 60^{\circ}+\tan ^2 45^{\circ}$
$\tan 30^{\circ}=\frac{1}{\sqrt{3}}$
$\tan 60^{\circ}=\sqrt{3}$
$\tan 45^{\circ}=1$
$\tan ^2 30^{\circ}+\tan ^2 60^{\circ}+\tan ^2 45^{\circ}$
$=\left(\frac{1}{\sqrt{3}}\right)^2+(\sqrt{3})^2+1$
$=\frac{1}{3}+3+1$
$=\frac{1+9+3}{3}$
$=\frac{13}{3} .$
$\tan 30^{\circ}=\frac{1}{\sqrt{3}}$
$\tan 60^{\circ}=\sqrt{3}$
$\tan 45^{\circ}=1$
$\tan ^2 30^{\circ}+\tan ^2 60^{\circ}+\tan ^2 45^{\circ}$
$=\left(\frac{1}{\sqrt{3}}\right)^2+(\sqrt{3})^2+1$
$=\frac{1}{3}+3+1$
$=\frac{1+9+3}{3}$
$=\frac{13}{3} .$
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