Question

The value of the expression $\tan {{30}^{\circ }}\tan {{60}^{\circ }}$ is equal to?

Answer 1

Hint: In trigonometry, we have some formulas which relate tan and cot functions. One of this formula is for complementary angles i.e. $\tan x=\cot \left( 90-x \right)$. The other formula which relates the tan and cot function is $\cot x=\dfrac{1}{\tan x}$. Using these formulas, we can solve this question.

“Complete step-by-step answer:”
Before proceeding with the question, we must know all the formulas that will be required to solve this question.
In trigonometry, we have a formula that relates the two trigonometric functions, tan and cot on any angle x in degrees. This formula is,
$\tan x=\cot \left( 90-x \right)$ . . . . . . . . . . . . . . . . . . . . (1)
Also, we have one more formula which can relate tan and cot functions. That formula is,
$\cot x=\dfrac{1}{\tan x}$ . . . . . . . . . . . . . . . (2)
In this question, we have to find the value of $\tan {{30}^{\circ }}\tan {{60}^{\circ }}$. Using formula (1), we can write $\tan 60=\cot \left( 90-60 \right)$.
$\Rightarrow \tan 60=\cot 30$
Also, using formula (2), we can write $\cot 30=\dfrac{1}{\tan 30}$. So, substituting $\tan 60=\dfrac{1}{\tan 30}$ in the expression given in the question, we get,
$\begin{align}
  & \tan {{30}^{\circ }}\tan {{60}^{\circ }}=\tan {{30}^{\circ }}\times \dfrac{1}{\tan {{30}^{\circ }}} \\
 & \Rightarrow \tan {{30}^{\circ }}\tan {{60}^{\circ }}=1 \\
\end{align}$
Hence, the answer is 1.

Note: There is an alternate way to do this question. One can also do this question directly if he/she has remembered the value of $\tan {{30}^{\circ }}$ is equal to $\dfrac{1}{\sqrt{3}}$ and the value of $\tan {{60}^{\circ }}$ is equal to $\sqrt{3}$. Since ${{30}^{\circ }}$ and ${{60}^{\circ }}$ are standard angles in trigonometry, the value of all the trigonometric functions for these two angles are known.