Question

How do you simplify the expression $\dfrac{{\sec \theta }}{{\tan \theta }}$ ?

Answer 1

Hint: To simplify the given trigonometric ratio, we use some standard properties and formulas of trigonometry. We will first convert the \[\sec \theta \] and \[\tan \theta \] in sin and cos terms and then try to simplify that ratio.

Complete step-by-step answer:
The given expression is $\dfrac{{\sec \theta }}{{\tan \theta }}$
We first rewrite \[\sec \theta \] in the form of cosine function as $\sec \theta = \dfrac{1}{{\cos \theta }}$,
$ =\dfrac{{\dfrac{1}{{\cos \theta }}}}{{\tan \theta }}$
Now we rewrite \[\tan \theta \] in the form of sine and cosine function as $\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}$,
$= \dfrac{{\dfrac{1}{{\cos \theta }}}}{{\dfrac{{\sin \theta }}{{\cos \theta }}}}$
Cancelling $\cos \theta $ from the denominators,
 $= \dfrac{1}{{\sin \theta }}$
We know that, $\dfrac{1}{{\sin \theta }} = \cos ec\theta $,
$= \cos ec\theta $
Therefore, we get $\dfrac{{\sec \theta }}{{\tan \theta }} = \cos ec\theta $
Hence, the expression is simplified.

Note: The division $\dfrac{{\dfrac{1}{{\cos \theta }}}}{{\dfrac{{\sin \theta }}{{\cos \theta }}}}$ can be alternatively done by using the division sign.
$ \Rightarrow \dfrac{1}{{\cos \theta }} \div \dfrac{{\sin \theta }}{{\cos \theta }}$
Removing the division sign and replacing it with a multiplication sign , we reverse the second term as follows,
$ \Rightarrow \dfrac{1}{{\cos \theta }} \times \dfrac{{\cos \theta }}{{\sin \theta }}$
Cancelling $\cos \theta $ terms from numerator and denominator,
$ \Rightarrow \dfrac{1}{1} \times \dfrac{1}{{\sin \theta }}$
And now continuing the same steps as above we get the simplified expression.