Question

Choose the correct option and justify your choice:
$\dfrac{2\tan {{30}^{\circ }}}{1+{{\tan }^{2}}{{30}^{\circ }}}$

a) $\sin {{60}^{\circ }}$
b) $\cos {{60}^{\circ }}$
c) $tan{{60}^{\circ }}$
d) $\sin {{30}^{\circ }}$

Answer 1

Hint: We know that value of $\tan {{30}^{\circ }}=\dfrac{1}{\sqrt{3}}$ so substituting this value of tan 30° in the given expression $\dfrac{2\tan {{30}^{\circ }}}{1+{{\tan }^{2}}{{30}^{\circ }}}$and evaluate then solve all the options given in question and compare the result of options with the result of evaluated expression.

Complete step-by-step answer:

The expression given in the question that we have to evaluate is:

$\dfrac{2\tan {{30}^{\circ }}}{1+{{\tan }^{2}}{{30}^{\circ }}}$

If we have the value of tan 30° in the above expression then we can easily evaluate this expression.

So, from the table of trigonometric values we know that $\tan {{30}^{\circ }}=\dfrac{1}{\sqrt{3}}$. Substituting this value of tan 30° in the given expression we get,

$\begin{align}

  & \dfrac{2\tan {{30}^{\circ }}}{1+{{\tan }^{2}}{{30}^{\circ }}} \\

 & =\dfrac{2\left( \dfrac{1}{\sqrt{3}} \right)}{1+\left( \dfrac{1}{3} \right)}=\dfrac{\dfrac{2}{\sqrt{3}}}{\dfrac{4}{3}}=\dfrac{\sqrt{3}}{2} \\

\end{align}$

The evaluation of the given expression$\dfrac{2\tan {{30}^{\circ }}}{1+{{\tan }^{2}}{{30}^{\circ }}}$is$\dfrac{\sqrt{3}}{2}$.

Now, we are going to check the options given in the question.

a) $\sin {{60}^{\circ }}=\dfrac{\sqrt{3}}{2}$

This option has the same value as of the given expression.

b) $\cos {{60}^{\circ }}=\dfrac{1}{2}$

This option has not the same value as of the given expression.

c) $\tan {{60}^{\circ }}=\sqrt{3}$

This option has not the same value as of the given expression.

d) $\sin {{30}^{\circ }}=\dfrac{1}{2}$

This option has not the same value as of the given expression.

Hence, the correct option is (a).

Note: The alternate way of solving the problem demands better understanding of trigonometric identities.

The expression given in the question is:

$\dfrac{2\tan {{30}^{\circ }}}{1+{{\tan }^{2}}{{30}^{\circ }}}$


If you can recall the identity of the double angle of sine which is:


$\sin 2\theta =\dfrac{2\tan \theta }{1+{{\tan }^{2}}\theta }$


Comparing the given expression with the above identity, the value of θ is equal to 30°. So, the given expression is resolved to:

$\dfrac{2\tan {{30}^{\circ }}}{1+{{\tan }^{2}}{{30}^{\circ }}}=\sin {{60}^{\circ }}$

Now, you can check the options and see which option is equal to sin 60°.

Hence, the correct option is (a).