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.

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.

Standard Identities

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 + tan⁡²θ = sec⁡²θ
• 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 + tan⁡²θ = sec⁡²θ

tan²θ = (3/4)² = 9/16

sec⁡²θ = 1 + tan⁡²θ

= 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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Frequently Asked Questions

What are Standard Identities?

Standard identities are well-known algebraic formulas used to simplify mathematical expressions.

What do you mean by Identities?

Identity can be defined as the general forms to do complex problems easy.

What are the Uses of Identity?

The basic use of the identity is to make the mathematical problem simpler and easier to solve.

Is Every Equation an Identity?

No, but every identity is a equation.

How many Standard Identities are there?

There are 3 standard identities which are as follows:

• (a+b)² = a²+2ab+b²
• (a–b)² = a²–2ab+b²
• (a+b)(a–b) = a²–b²