Question

How do you simplify $\dfrac{{\tan x}}{{1 - {{\cos }^2}x}}$?

Answer 1

In the given question, we have been given an expression. This expression has two trigonometric functions – one in the numerator and one in the denominator. We have to simplify the value of the trigonometric functions as a whole in the expression. We know that all the trigonometric functions can be represented in a combination of sine and cosine and that is how we will simplify each of the trigonometric functions, and then combine them both to get a single answer for the whole expression.

Complete step-by-step answer:
The given expression is $p = \dfrac{{\tan x}}{{1 - {{\cos }^2}x}}$.
Now, we know that:
$\tan x = \dfrac{{\sin x}}{{\cos x}}$
and
$1 - {\cos ^2}x = {\sin ^2}x$
Hence, $p = \dfrac{{\dfrac{{\sin x}}{{\cos x}}}}{{{{\sin }^2}x}} = \dfrac{{{{\sin x}}}}{{\cos x}} \times \dfrac{1}{{{{{{\sin }^2}x}}\sin x}} = \dfrac{1}{{\sin x\cos x}}$
We know, $\sin 2x = 2\sin x\cos x$
Thus, $p = \dfrac{1}{{\dfrac{1}{2}\sin 2x}} = \dfrac{2}{{\sin 2x}}$
We know, $\csc x = \dfrac{1}{{\sin x}}$
Hence, $p = 2\csc 2x$

Additional Information:
We got the answer to this expression containing the two trigonometric functions by substituting the values of the cosine and tangent as the combination of values of sine and cosine. Perhaps if we want to simplify any expression containing the trigonometric functions, we can use these two to get to the answer.

Note: To do any kind of simplification of trigonometric functions, we can just simplify them into sine and cosine and then combine them and then solve them. We just need to remember all the basic trigonometric identities.