Question

The value of $\dfrac{2\tan {{30}^{\circ }}}{1+{{\tan }^{2}}{{30}^{\circ }}}$ is equal to:
(a) $\sin {{60}^{\circ }}$
(b) $\tan {{60}^{\circ }}$
(c) $\cos {{60}^{\circ }}$
(d) $\sin {{30}^{\circ }}$

Answer 1

Hint: In the expression given in the question, if we know the value of $\tan {{30}^{\circ }}$ then we can solve the given expression easily. From the trigonometric function values, we know that the value of $\tan {{30}^{\circ }}$ is equal to $\dfrac{1}{\sqrt{3}}$ then substitute this value in the given expression and solve.

Complete step-by-step answer:
The expression given in the question is:
$\dfrac{2\tan {{30}^{\circ }}}{1+{{\tan }^{2}}{{30}^{\circ }}}$
If we know the value of $\tan {{30}^{\circ }}$ then we can easily solve the above problem.
From the trigonometric function values, the value of $\tan {{30}^{\circ }}$ is equal to $\dfrac{1}{\sqrt{3}}$ . Substituting this value of $\tan {{30}^{\circ }}$ in the given expression we get,
$\begin{aligned} & \dfrac{2\left( \dfrac{1}{\sqrt{3}} \right)}{1+{{\left( \dfrac{1}{\sqrt{3}} \right)}^{2}}} \\\ & =\dfrac{\dfrac{2}{\sqrt{3}}}{\dfrac{4}{3}}=\dfrac{\dfrac{1}{\sqrt{3}}}{\dfrac{2}{3}}=\dfrac{1}{\sqrt3}\times\dfrac{3}{2}=\dfrac{\sqrt{3}}{2} \\\ \end{aligned}$
From the above simplification of the given expression we have got the value is equal to $\dfrac{\sqrt{3}}{2}$ .
Now, solve all the options given in the question and compare the result of the options with the answer given by solving the given expression.
(a) $\sin {{60}^{\circ }}=\dfrac{\sqrt{3}}{2}$
The above option has the same answer as that of the expression given in the question.
(b) $\tan {{60}^{\circ }}=\sqrt{3}$
The answer in the above option is not equal to that of the given expression.
(c) $\cos {{60}^{\circ }}=\dfrac{1}{2}$
The answer in the above option is not equal to that of the given expression.
(d) $\sin {{30}^{\circ }}=\dfrac{1}{2}$
The answer in the above option is not equal to that of the given expression.
From the above discussion, we can see that only option (a) has the same value as that of the given expression.
Hence, the correct option is (a).

Note: The alternative of the above method is discussed below:
The expression given in the question is:
$\dfrac{2\tan {{30}^{\circ }}}{1+{{\tan }^{2}}{{30}^{\circ }}}$
If you look carefully the above expression, you will find that the above expression is the expansion of the identity:
$\sin \left( 2\left( {{30}^{\circ }} \right) \right)=\dfrac{2\tan {{30}^{\circ }}}{1+{{\tan }^{2}}{{30}^{\circ }}}$
So, the given expression is equal to:
$\dfrac{2\tan {{30}^{\circ }}}{1+{{\tan }^{2}}{{30}^{\circ }}}=\sin {{60}^{\circ }}$
As we have seen from above, the simplification of the given expression yields the value $\sin {{60}^{0}}$ so the correct option is (a).