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If \( \theta=30^{\circ} \), verify that:
\( \tan 2 \theta=\frac{2 \tan \theta}{1-\tan ^{2} \theta} \)
Given:
\( \theta=30^{\circ} \)
To do:
We have to verify that \( \tan 2 \theta=\frac{2 \tan \theta}{1-\tan ^{2} \theta} \).
Solution:
\( \tan 2 \theta=\frac{2 \tan \theta}{1-\tan ^{2} \theta} \)
This implies,
\( \tan 2(30^{\circ})=\frac{2 \tan 30^{\circ}}{1-\tan ^{2} 30^{\circ}} \)
\( \tan 60^{\circ}=\frac{2 \tan 30^{\circ}}{1-\tan ^{2} 30^{\circ}} \)
We know that,
$\tan 60^{\circ}=\sqrt3$
$\tan 30^{\circ}=\frac{1}{\sqrt3}$
Let us consider LHS,
$\tan 2 \theta=\tan 60^{\circ}$
$=\sqrt3$
Let us consider RHS,
$\frac{2 \tan \theta}{1-\tan ^{2} \theta}=\frac{2 \tan 30^{\circ}}{1-\tan ^{2} 30^{\circ}}$
$=\frac{2\left(\frac{1}{\sqrt{3}}\right)}{1-\left(\frac{1}{\sqrt{3}}\right)^{2}}$
$=\frac{\frac{2}{\sqrt{3}}}{1-\frac{1}{3}}$
$=\frac{\frac{2}{\sqrt{3}}}{\frac{3-1}{3}}$
$=\frac{\frac{2}{\sqrt{3}}}{\frac{2}{3}}$
$=\frac{2}{\sqrt{3}} \times \frac{3}{2}$
$=\frac{\sqrt{3} \times \sqrt{3}}{\sqrt{3}}$
$=\sqrt{3}$
LHS $=$ RHS
Hence proved.