Question

Why is $1 - {\cos ^3}x = (1 - \cos x)(1 - 2\cos x + {\cos ^2}x)$ a conditional equation?

Answer 1

A conditional equation is an equation that is true for some value or values of the variable, but not true for other values of the variable. For example: The equation $2x - 7 = 9$ is conditional because it is only true for $x = 8$.

Complete step by step answer:
Given equation $1 - {\cos ^3}x = (1 - \cos x)(1 - 2\cos x + {\cos ^2}x)$
On solving the right side of the equation. We get,
$1 - {\cos ^3}x = 1(1 - 2\cos x + {\cos ^2}x) - \cos x(1 - 2\cos x + {\cos ^2}x)$
$\Rightarrow 1 - {\cos ^3}x = 1 - 2\cos x + {\cos ^2}x - \cos x + 2{\cos ^2}x - {\cos ^3}x$
Cancelling out the equal terms. We get,
$3{\cos ^2}x - 3\cos x = 0$
Taking out $3$ common from both the terms. We get,
$3({\cos ^2}x - \cos x) = 0$

Shifting $3$ to the right side of the equation. We get,
${\cos ^2}x - \cos x = 0$
Now we put the different value of $x$ to check that the given equation is conditional or not.
Let $x = 0$. Put $x = 0$ in the left side of the equation
${\cos ^2}(0) - \cos (0)$ $= 0 - 0 = 0$
Hence, $x = 0$ satisfies the given equation.
Let $x = 1$. Put $x = 1$ in the left side of the equation
${\cos ^2}(1) - \cos (1)$ $=$ ${(0.99)^2} - 0.99$
$\therefore {\cos ^2}(1) - \cos (1)= 0.9801 - 0.99 = - 0.0099$
Hence, $x = 1$ does not satisfy the equation.

Hence, the equation $1 - {\cos ^3}x = (1 - \cos x)(1 - 2\cos x + {\cos ^2}x)$ is valid and only true for $x = 0$. So, this is a conditional equation.

Note: Conditional and identity equations are ways in which numbers associate with each other. When an equation is true for every value of the variable, then the equation is called an identity equation. Whereas when an equation is false for at least one value, it is called a conditional equation. If solving a linear equation leads to a true statement such as $0 = 0$, the equation is an identity. Its solution set is all real numbers. If solving a linear equation leads to a single solution such as $x = 8$, the equation is conditional.