Question

How do you prove \[\dfrac{{1 - \cos x}}{{\sin x}} = \dfrac{{\sin x}}{{1 + \cos x}}\]?

Answer 1

Hint: Here, we will first rationalize the LHS by multiplying the conjugate of the numerator of the faction to both the numerator and denominator. Then we will simplify it using the algebraic and trigonometric identities. We will solve the expression further by canceling out the same terms to prove that the given LHS is equal to the RHS.

Formula Used:
1. \[\left( {a - b} \right)\left( {a + b} \right) = {a^2} - {b^2}\]
2. \[{\sin ^2}x + {\cos ^2}x = 1\]

Complete step-by-step answer:
We will first take into consideration the left-hand side of the given trigonometric equation.
LHS \[ = \dfrac{{1 - \cos x}}{{\sin x}}\]
Now, we will do reverse rationalizing.
Usually, we rationalize a fraction with the help of the denominator but in this case, we will rationalize with the help of the numerator.
So, by multiplying and dividing the LHS by \[1 + \cos x\], we get
\[ \Rightarrow \] LHS \[ = \dfrac{{1 - \cos x}}{{\sin x}} \times \dfrac{{1 + \cos x}}{{1 + \cos x}}\]
\[ \Rightarrow \] LHS \[ = \dfrac{{\left( {1 - \cos x} \right)\left( {1 + \cos x} \right)}}{{\sin x\left( {1 + \cos x} \right)}}\]
Using the identity \[\left( {a - b} \right)\left( {a + b} \right) = {a^2} - {b^2}\] in the numerator, we get
\[ \Rightarrow \] LHS \[ = \dfrac{{1 - {{\cos }^2}x}}{{\sin x\left( {1 + \cos x} \right)}}\]
We know that, \[{\sin ^2}x + {\cos ^2}x = 1\] or \[1 - {\cos ^2}x = {\sin ^2}x\].
Thus, substituting this in the numerator, we get
\[ \Rightarrow \] LHS\[ = \dfrac{{{{\sin }^2}x}}{{\sin x\left( {1 + \cos x} \right)}}\]
Canceling out the same terms from the numerator and the denominator, we get
\[ \Rightarrow \] LHS \[ = \dfrac{{\sin x}}{{1 + \cos x}} = \] RHS
Therefore, LHS \[ = \] RHS

So, \[\dfrac{{1 - \cos x}}{{\sin x}} = \dfrac{{\sin x}}{{1 + \cos x}}\]
Hence, proved

Note:
In this question, we have used trigonometry. 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.