Question

Simplify the trigonometric equation $(1 + \cos x)(1 - \cos x)$?

Answer 1

For solving such a question you need to be careful with your multiplication, since these expressions are not a very big deal you can easily go through it. For simplification purposes you may need to know the basic trigonometric identity.

Formulae Used: ${\sin ^2}x + {\cos ^2}x = 1$ is used here.

Complete step by step answer:
Given equation $(1 + \cos x)(1 - \cos x)$
Expanding it we get:

$$\Rightarrow 1(1 - \cos x) + \cos x(1 - \cos x) \\\ \Rightarrow 1 - \cos x + \cos x - {\cos ^2}x \\\ \Rightarrow 1 - {\cos ^2}x \\\ \Rightarrow {\sin ^2}x \\\$$

Note: In the expansion of two bracket in product the rule is to multiply the $1^{st}$ term or the first bracket with the whole next bracket then put the sign of the $2^{nd}$ term then again multiply the $2^{nd}$ term with the whole next bracket, if more terms are there in the first bracket then continue the same process for the rest of the terms. After that just simply do your multiplication and get the results. While multiplication you should be so clear with the signs of the terms, because these are the silly mistakes that happen with such questions.
Talking about the trigonometric identity used here, it is a very basic identity and has its proof also, you can prove it on any angle you wish you will get the results true. For example for an angle of 45 degree if we use the above trigonometric identity then we will get L.H.S = R.H.S.