Question

How do you simplify $\dfrac{{\tan \theta }}{{\cot \theta }}$?

Answer 1

Here, we will use different trigonometric relations to find the answer to the given question. We will first express the numerator and denominator in terms of sine and cosine functions. Then we will simplify it using basic mathematical operations. We will again use the relation between tangent, sine, and cosine function to get the required answer.

Formula Used:
We will use the following formulas:
$\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}$
$\cot \theta = \dfrac{{\cos \theta }}{{\sin \theta }}$

Complete step-by-step answer:
In order to simplify $\dfrac{{\tan \theta }}{{\cot \theta }}$, we will use the relationship between various trigonometric functions.
We know that $\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}$ and $\cot \theta = \dfrac{{\cos \theta }}{{\sin \theta }}$.
Now, substituting these values in the given expression, we get,
$\dfrac{{\tan \theta }}{{\cot \theta }} = \dfrac{{\dfrac{{\sin \theta }}{{\cos \theta }}}}{{\dfrac{{\cos \theta }}{{\sin \theta }}}}$
Rewriting the equation, we get
$\Rightarrow \dfrac{{\tan \theta }}{{\cot \theta }} = \dfrac{{\sin \theta }}{{\cos \theta }} \times \dfrac{{\sin \theta }}{{\cos \theta }}$
Multiplying the terms, we get
$\Rightarrow \dfrac{{\tan \theta }}{{\cot \theta }} = {\left( {\dfrac{{\sin \theta }}{{\cos \theta }}} \right)^2}$
Again substituting $\dfrac{{\sin \theta }}{{\cos \theta }} = \tan \theta$, we get,

$\Rightarrow \dfrac{{\tan \theta }}{{\cot \theta }} = {\tan ^2}\theta$.

Thus, this is the required answer.

Additional information:
Trigonometry is a branch of mathematics that helps us to study the relationship between the sides and the angles of a triangle. In practical life, trigonometry is used by cartographers (to make maps). It is also used by the aviation and naval industries. In fact, trigonometry is even used by Astronomers to find the distance between two stars. Hence, it has an important role to play in everyday life. The three most common trigonometric functions are the tangent function, the sine, and the cosine function. In simple terms, they are written as ‘sin’, ‘cos’, and ‘tan’. Hence, trigonometry is not just a chapter to study, in fact, it is being used in everyday life.

Note: An alternate way of solving this question is to use the relation that:
We know that $\cot \theta = \dfrac{1}{{\tan \theta }}$.
Hence, substituting this in $\dfrac{{\tan \theta }}{{\cot \theta }}$, we get,
\dfrac{{\tan \theta }}{{\cot \theta }} = \dfrac{{\tan \theta }}{{\dfrac{1}{{\tan \theta }}}} \\\ \Rightarrow \dfrac{{\tan \theta }}{{\cot \theta }} = \tan \theta \times \tan \theta \\\
Multiplying the terms, we get
$\Rightarrow \dfrac{{\tan \theta }}{{\cot \theta }} = {\tan ^2}\theta$
Therefore, $\dfrac{{\tan \theta }}{{\cot \theta }} = {\tan ^2}\theta$
Thus, this is the required answer.