Question

How do you simplify \[\left( {1 - \cos x} \right)\left( {1 + \cos x} \right)\]?

Answer 1

Hint: Here, we will multiply the given trigonometric expression by using the FOIL method. Then by using the trigonometric identity, we will simplify the given trigonometric expression. FOIL method is a method of multiplying the binomials by multiplying the first terms, then the outer terms, then the inner terms and at last the last terms. It is possible to combine like terms.

Formula Used:
Trigonometric identity: \[{\sin ^2}x + {\cos ^2}x = 1\]

Complete Step by Step Solution:
We are given a trigonometric expression \[\left( {1 - \cos x} \right)\left( {1 + \cos x} \right)\].
Now, we will multiply the terms in the given trigonometric expression by using the FOIL method.
\[ \Rightarrow \left( {1 - \cos x} \right)\left( {1 + \cos x} \right) = 1\left( {1 + \cos x} \right) - \cos x\left( {1 + \cos x} \right)\]
Now, by multiplying the terms with the terms of another trigonometric expression, we get
\[ \Rightarrow \left( {1 - \cos x} \right)\left( {1 + \cos x} \right) = 1 + \cos x - \cos x - {\cos ^2}x\]
Adding and subtracting the like terms, we get
\[ \Rightarrow \left( {1 - \cos x} \right)\left( {1 + \cos x} \right) = 1 - {\cos ^2}x\]
From the identity \[{\sin ^2}x + {\cos ^2}x = 1\] , we can say that
\[1 - {\cos ^2}x = {\sin ^2}x\]
Now by using this value in above equation, we get
\[ \Rightarrow \left( {1 - \cos x} \right)\left( {1 + \cos x} \right) = {\sin ^2}x\].

Therefore, the value of \[\left( {1 - \cos x} \right)\left( {1 + \cos x} \right)\] is \[{\sin ^2}x\].

Note:
We know that Trigonometric Equation is defined as an equation involving trigonometric ratios. Trigonometric identity is an equation that is always true for all the variables. We should know that we have many trigonometric identities that are related to all the other trigonometric equations. Trigonometric Ratios of a Particular angle are the ratios of the sides of a right-angled triangle with respect to any of its acute angle. Trigonometric Ratios are used to find the relationships between the sides of a right-angle triangle. We should remember that all the trigonometric ratios can be written in the form of a basic Trigonometric Ratio of sine and cosine.