1.3.3
Proving an Identity
How do you prove an identity?
How do you prove an identity?
To prove an identity is always true, you need to show one side is the same as the other. The identity symbol is ≡, which means 'equivalent'.
![Illustrative background for Identity](https://image-v2.cdn.app.senecalearning.com/2018-04/e2b9bb9f-f1f5-4e67-91e4-cfe3bdf35903/Screen%20Shot%202018-04-12%20at%2009.59.05,h_400,q_80,w_640.png)
![Illustrative background for Identity ?? "content](https://image-v2.cdn.app.senecalearning.com/2018-04/e2b9bb9f-f1f5-4e67-91e4-cfe3bdf35903/Screen%20Shot%202018-04-12%20at%2009.59.05,h_400,q_80,w_640.png)
Identity
Identity
- Prove that .
![Illustrative background for Left-hand side](https://image-v2.cdn.app.senecalearning.com/2018-04/bd83c618-e78e-4678-8b2b-ab2fd1c00799/Screen%20Shot%202018-04-12%20at%2009.59.09,h_400,q_80,w_640.png)
![Illustrative background for Left-hand side ?? "content](https://image-v2.cdn.app.senecalearning.com/2018-04/bd83c618-e78e-4678-8b2b-ab2fd1c00799/Screen%20Shot%202018-04-12%20at%2009.59.09,h_400,q_80,w_640.png)
Left-hand side
Left-hand side
- Consider the left-hand side of the identity .
![Illustrative background for Expand brackets](https://image-v2.cdn.app.senecalearning.com/2018-04/44032706-b0e8-48d7-a6ec-03879658c217/Screen%20Shot%202018-04-12%20at%2009.59.19,h_400,q_80,w_640.png)
![Illustrative background for Expand brackets ?? "content](https://image-v2.cdn.app.senecalearning.com/2018-04/44032706-b0e8-48d7-a6ec-03879658c217/Screen%20Shot%202018-04-12%20at%2009.59.19,h_400,q_80,w_640.png)
Expand brackets
Expand brackets
- First, expand the brackets using FOIL. This step is known to always be true.
![Illustrative background for Collect terms](https://image-v2.cdn.app.senecalearning.com/2018-04/aa144936-ab60-4464-acda-bb61fb756bf2/Screen%20Shot%202018-04-12%20at%2009.59.24,h_400,q_80,w_640.png)
![Illustrative background for Collect terms ?? "content](https://image-v2.cdn.app.senecalearning.com/2018-04/aa144936-ab60-4464-acda-bb61fb756bf2/Screen%20Shot%202018-04-12%20at%2009.59.24,h_400,q_80,w_640.png)
Collect terms
Collect terms
- Collect all the terms of equal powers of n to give:
![Illustrative background for Right-hand side](https://image-v2.cdn.app.senecalearning.com/2018-04/72665d91-c4f8-4bf3-908b-8b58f668c09f/Screen%20Shot%202018-04-12%20at%2009.59.29,h_400,q_80,w_640.png)
![Illustrative background for Right-hand side ?? "content](https://image-v2.cdn.app.senecalearning.com/2018-04/72665d91-c4f8-4bf3-908b-8b58f668c09f/Screen%20Shot%202018-04-12%20at%2009.59.29,h_400,q_80,w_640.png)
Right-hand side
Right-hand side
- Comparing to the right-hand side of the identity, we see that they are always equal.
![Illustrative background for Final statement](https://image-v2.cdn.app.senecalearning.com/2020-04/f1643cbe-5f3c-4db6-87cc-17cc96490a7f/identity-statement,h_400,q_80,w_640.png)
![Illustrative background for Final statement ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-04/f1643cbe-5f3c-4db6-87cc-17cc96490a7f/identity-statement,h_400,q_80,w_640.png)
Final statement
Final statement
- The final statement is that the identity holds:
- So
1Proof
1.1Types of Numbers
1.2Notation
2Algebra & Functions
2.1Powers & Roots
2.2Quadratic Equations
2.3Inequalities
2.4Polynomials
2.5Graphs
2.7Transformation of Graphs
3Coordinate Geometry
3.1Straight Lines
3.2Circles
3.2.1Equations of Circles centred at Origin
3.2.2Finding the Centre & Radius
3.2.3Equation of a Tangent
3.2.4Circle Theorems - Perpendicular Bisector
3.2.5Circle Theorems - Angle at the Centre
3.2.6Circle Theorems - Angle at a Semi-Circle
3.2.7Equation of a Perpendicular Bisector
3.2.8Equation of a Circumcircle
3.2.9Circumcircle of a Right-angled Triangle
3.3Parametric Equations (A2 only)
4Sequences & Series
4.1Binomial Expansion
5Trigonometry
5.2Trigonometric Functions
5.3Triangle Rules
6Exponentials & Logarithms
6.1Exponentials & Logarithms
7Differentiation
7.1Derivatives
7.2Graphs & Differentiation
7.3Differentiation With Trigonometry and Exponentials
7.4Rules of Differetiation (A2 only)
7.5Parametric & Implicit Differentiation
8Integration
8.1Integration
9Numerical Methods
9.1Finding Solutions
9.2Finding the Area
10Vectors
10.12D Vectors
10.23D Vectors
10.3Vector Proofs
Jump to other topics
1Proof
1.1Types of Numbers
1.2Notation
2Algebra & Functions
2.1Powers & Roots
2.2Quadratic Equations
2.3Inequalities
2.4Polynomials
2.5Graphs
2.7Transformation of Graphs
3Coordinate Geometry
3.1Straight Lines
3.2Circles
3.2.1Equations of Circles centred at Origin
3.2.2Finding the Centre & Radius
3.2.3Equation of a Tangent
3.2.4Circle Theorems - Perpendicular Bisector
3.2.5Circle Theorems - Angle at the Centre
3.2.6Circle Theorems - Angle at a Semi-Circle
3.2.7Equation of a Perpendicular Bisector
3.2.8Equation of a Circumcircle
3.2.9Circumcircle of a Right-angled Triangle
3.3Parametric Equations (A2 only)
4Sequences & Series
4.1Binomial Expansion
5Trigonometry
5.2Trigonometric Functions
5.3Triangle Rules
6Exponentials & Logarithms
6.1Exponentials & Logarithms
7Differentiation
7.1Derivatives
7.2Graphs & Differentiation
7.3Differentiation With Trigonometry and Exponentials
7.4Rules of Differetiation (A2 only)
7.5Parametric & Implicit Differentiation
8Integration
8.1Integration
9Numerical Methods
9.1Finding Solutions
9.2Finding the Area
10Vectors
10.12D Vectors
10.23D Vectors
10.3Vector Proofs
![Go student ad image](/en-GB/revision-notes/_next/image?url=%2Fen-GB%2Frevision-notes%2Fimages%2Fgo-student-uk-ad.jpg&w=640&q=100)
Unlock your full potential with GoStudent tutoring
Affordable 1:1 tutoring from the comfort of your home
Tutors are matched to your specific learning needs
30+ school subjects covered