Identities – Study Notes for ITI Admission Test
(Mathematics – Algebra & Basic Trigonometry)
1. What is an Identity?
An identity is an equation that holds true for all values of the variables involved (within the domain of the expression).
Example: ((a+b)^2 = a^2 + 2ab + b^2) is true for every real (or complex) numbers (a) and (b).
Use identities to
- simplify expressions
- factorise polynomials
- solve equations quickly
- check work or prove other results
2. Key Algebraic Identities
| Symbol | Identity | When to Use |
|---|---|---|
| Squares | ((a+b)^2 = a^2 + 2ab + b^2) | Expand a binomial squared |
| ((a-b)^2 = a^2 - 2ab + b^2) | Expand a binomial squared | |
| Difference of Squares | ((a+b)(a-b) = a^2 - b^2) | Factorise (a^2 - b^2) |
| Cubes | ((a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3) | Expand a binomial cubed |
| ((a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3) | Expand a binomial cubed | |
| Sum & Difference of Cubes | (a^3+b^3 = (a+b)(a^2 - ab + b^2)) | Factorise (a^3 + b^3) |
| (a^3-b^3 = (a-b)(a^2 + ab + b^2)) | Factorise (a^3 - b^3) | |
| Factorised Forms | (a^2-b^2 = (a-b)(a+b)) | Quick factorisation |
| (a^4-b^4 = (a^2-b^2)(a^2+b^2) = (a-b)(a+b)(a^2+b^2)) | Factorise difference of fourth powers | |
| Geometric Series (finite) | (1 + r + r^2 + \cdots + r^{n-1} = \dfrac{r^n-1}{r-1}) (for (r\neq 1)) | Sum of a geometric progression |
| Triangular Number | (1+2+3+\cdots+n = \dfrac{n(n+1)}{2}) | Sum of first (n) natural numbers |
| Square of Sum | ((a+b+c)^2 = a^2+b^2+c^2+2(ab+ac+bc)) | Expand a triple binomial squared |
Tip: Memorise the “difference of squares” and the “sum/difference of cubes” identities first; they appear most often in exam problems.
3. Basic Trigonometric Identities (Optional)
| Symbol | Identity | Use |
|---|---|---|
| Pythagorean | (\sin^2\theta + \cos^2\theta = 1) | Simplify trigonometric expressions |
| Angle Addition (Cosine) | (\cos(\alpha+\beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta) | Expand or simplify (\cos) of a sum |
| Angle Addition (Sine) | (\sin(\alpha+\beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta) | Expand or simplify (\sin) of a sum |
| Double Angle | (\cos 2\theta = \cos^2\theta - \sin^2\theta = 2\cos^2\theta-1 = 1-2\sin^2\theta) | Reduce powers of sine/cosine |
| Product to Sum | (\sin A \sin B = \frac{1}{2}[\cos(A-B)-\cos(A+B)]) | Convert products to sums |
Note: The IIT/ITI exam rarely tests advanced trig identities; focus on the algebraic identities above.
4. Worked Example
Problem
Simplify the expression:
[
\frac{(x^3-27)+(x^3+27)}{(x^2-9)} \quad \text{for } x\neq \pm 3
]
Solution
-
Factor the numerator
- (x^3-27 = x^3-3^3 = (x-3)(x^2+3x+9))
- (x^3+27 = x^3+3^3 = (x+3)(x^2-3x+9))
So, [ (x^3-27)+(x^3+27) = (x-3)(x^2+3x+9)+(x+3)(x^2-3x