Standard Identities are an important topic in Algebra that are used to solve various problems. They help in simplifying calculations, solving equations, and transforming expressions.
In this article, we will learn about some special identities that are very important in every part of mathematics. Also, we practice the types of questions related to these identities.
Table of Content
What are Standard Identities?
Standard identities are algebraic equations that are true for all values of the variables involved. These identities are fundamental tools in algebra, trigonometry, and other areas of mathematics, allowing for simplification and manipulation of expressions.
Algebraic Identities
Various Algebraic Identities are:
- (a+b)2 = a2+2ab+b2
- (a–b)2 = a2–2ab+b2
- (a+b)(a–b) = a2–b2
- (x+a)(x+b) = x2+(a+b)x+ab
- (a+b+c)2 = a2+b2+c2+2ab+2bc+2ac
- (a+b)3 = a3+b3+3ab(a+b)
- (a–b)3 = a3–b3–3ab(a–b)
- (a+b+c)(a2+b2+c2–ab–bc–ca) = a3+b3+c3–3abc
- (a + b) (a + c) (b + c) = (a + b + c) (ab + ac + bc) – abc
- a2 + b2 +c2 = (a + b + c)2 – 2(ab + ac + bc)
There are three standard identities which are as follows:
- (a+b)2 = a2+2ab+b2
- (a–b)2 = a2–2ab+b2
- (a+b)(a–b) = a2–b2
Trigonometric Identities
Some important Trigonometric Identities are:
- sin2θ + cos2θ = 1
- 1 + tan2θ = sec2θ
- 1 + cot2θ = csc2θ
Standard Identities Practice Questions with Solution
Problem 1: Find the product of (x + 1)2 using standard algebraic identities.
Solution:
Given equation is (x + 1)2
Using identity (a+b)2 = a2 + 2ab + b2
put a = x and b = 1
(x+1)2 = x2 + 2.x.1 + 12
(x+1)2 = x2 +2x +1
Problem 2: Factorise (x4 – 1) using standard algebraic identities.
Solution:
Given equation (x4 – 1)
(x4 – 1) = [ (x2)2 - 1]
Using identity: (a+b)(a–b) = a2 – b2
(x4 – 1) = (x2+1)(x2-1)
Now again, (x2– 1) = (x+1)(x-1)
Hence, (x4 – 1) = (x2+1) (x+1)(x-1)
Problem 3: Expand (3x – 4y)2 using identity.
Solution:
Given (3x – 4y)2
a = 3x and b = 4y
Using (a–b)2 = a2 – 2ab + b2
(3x – 4y)2 = (3x)2 - 2.3x.4y + (4y)2
(3x – 4y)2 = 9x2 - 24xy + 16y2
Problem 4: Solve using identities: 122 - 102
Solution:
Given 122 - 102
Using (a+b)(a–b) = a2–b2
a = 12 and b = 10
Now, 122 - 102 = (12+10)(12-10)
122 - 102 = 22 ×2 = 44
Problem 5: Expand (2x + 3y + z)2 using identities.
Solution:
Given (2x+3y+z)2
Using: (a+b+c)2 = a2+b2+c2+2ab+2bc+2ac
a = 2x, b = 3y and c = z
(2x+3y+z)2 = (2x)2+(3y)2+(z)2+2.2x.3y+2.3y.z+2.z.2x
(2x+3y+z)2 = 4x2 + 9y2 + z2 + 12xy + 6yz + 4zx
Problem 6: If sinθ = 3/5, find cosθ.
Solution:
Given, sinθ = 3/5
Using Identity: sin2θ + cos2θ = 1
sin2θ = (3/5)2 = 9/25
cos2θ = 1 - sin2θ
= 1 - 9/25 = 16/25
cos2θ = 16/25
cosθ = 4/5
Problem 7: If tanθ = 3/4, find cosθ.
Solution:
Given, tanθ = 3/4
Using Identity: 1 + tan2θ = sec2θ
tan2θ = (3/4)2 = 9/16
sec2θ = 1 + tan2θ
= 1 + 9/16 = 25/16
sec2θ = 25/16
secθ = 5/4
Standard Identities: Worksheet
Q1. Expand (3x – 4y)3 using standard algebraic identities.
Q2. Use the identities to calculate the value of (81)2.
Q3. Use the identities to calculate the value of (37)2.
Q4. Use the identities to calculate the value of 97 × 103.
Q5. Simplify (2z + 5)2 – (2z - 5)2.
Q6. Use the identities to calculate the value of 96 × 102.
Q7. Use the identities to calculate the value of 91 × 109.
Q8. Use the identities to calculate the value of (87)2.
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