An equation is a mathematical representation of a statement where two expressions are equal.
f(x) = g(x) is an equation where, and are expressions with independent variable x.
Note:- We can have more than one variable in an equation.
If the equation is true for every value of the variable(s) for which both expressions are defined (or domain of expressions) then it is an identity, otherwise it is a conditional equation.
Example 1: Equation and Identity
Explain why Set A of equations are conditional equations and equations of Set B are identities
Set A: 2x-1=5 x^{2}– 8x+7=0 x+y=0 x^{2}=-1
Set B: 3p +5p = 8p x^{2} – y^{2} = (x + y)(x – y) \(\frac{x^2 -1}{x + 1}\) = x -1 , x ≠ 1
Solution 1: Equation and Identity
Set A
Equation 2−1=5 is true only for x = 3
Equation 2−8+7=0 is true for x = 1 and x = 2
Equation +=0 is true for infinite value but not all values of x and y
Equation 2=−1 is not true for any real number
All the above equations in the Set A are conditional equation; they are not true for every value of independent variable in the domain.
Set B: 3p +5p = 8p x^{2} – y^{2} = (x + y)(x – y) \(\frac{x^2 -1}{x + 1}\) = x -1 , x ≠ 1
Set B of equations are identities as they are true for every value of variable(s) for which it is defined.
Note: Domain is the set of all possible values the independent variable can take.
Trigonometric identities involve trigonometric functions. List of fundamental trigonometric identities is provided in the beginning of the chapter and will be proved in the next section. The fundamental identities include:
• Three reciprocal identities
• Two quotient or ratio identity
• Three Pythagorean identities
• Three complementary angle or co-function identities
To prove an identity one side of an equation is shown as exactly equal to the other side of the equation. Start from more complicated side, transform and simplify expressions using algebraic skills and establish identities to get the other expression.
Student’s Activity
Is \(\sqrt{x^2}\)= an identity? Justify your answer or provide a counter example.