Question

If $\dfrac{2\tan 30{}^\circ }{1-{{\tan }^{2}}30{}^\circ }=\tan \theta$. Then the value of $\theta$ is:
(a) $60{}^\circ$
(b) $65{}^\circ$
(c) $45{}^\circ$
(d) $30{}^\circ$

Answer 1

Hint: Try to simplify the left-hand side of the equation that is given in the question by using the formula $\tan 2A=\dfrac{2\tan A}{1-{{\tan }^{2}}A}$, and other similar formulas.

Complete step-by-step answer:
We will now solve the left-hand side of the equation given in the question.
$\dfrac{2\tan 30{}^\circ }{1-{{\tan }^{2}}30{}^\circ }$
Now we know that $\tan 2A=\dfrac{2\tan A}{1-{{\tan }^{2}}A}$ . Therefore, our equation becomes:
$\tan \left( 2\times 30 \right){}^\circ$
$=\tan 60{}^\circ$
Therefore, we can say that:
$\tan \theta =\tan 60{}^\circ$
Now according to the rule of trigonometric equations: $\tan \beta =\tan \theta$ implies that $\theta =n\times 180{}^\circ +\beta$ .
$\therefore \theta =n\times 180{}^\circ +60{}^\circ$
Now, as n can be any integer, we let n=0.
$\theta =60{}^\circ$
Hence, the answer to the above question is option (a).

Note: Be careful, as $60{}^\circ$ is not the only possible value of $\theta$ . If we substitute the value of n in $\theta =n\times 180{}^\circ +60{}^\circ$ with different integers, the different values of $\theta$ we get are also acceptable values of $\theta$ . Also, we need to remember the properties related to complementary angles and trigonometric ratios.