#### Topics

##### Number Systems

##### Real Numbers

##### Algebra

##### Pair of Linear Equations in Two Variables

- Linear Equations in Two Variables
- Graphical Method of Solution of a Pair of Linear Equations
- Substitution Method
- Elimination Method
- Cross - Multiplication Method
- Equations Reducible to a Pair of Linear Equations in Two Variables
- Consistency of Pair of Linear Equations
- Inconsistency of Pair of Linear Equations
- Algebraic Conditions for Number of Solutions
- Simple Situational Problems
- Pair of Linear Equations in Two Variables
- Relation Between Co-efficient

##### Arithmetic Progressions

##### Quadratic Equations

- Quadratic Equations
- Solutions of Quadratic Equations by Factorization
- Solutions of Quadratic Equations by Completing the Square
- Nature of Roots
- Relationship Between Discriminant and Nature of Roots
- Situational Problems Based on Quadratic Equations Related to Day to Day Activities to Be Incorporated
- Quadratic Equations Examples and Solutions

##### Polynomials

##### Geometry

##### Circles

- Concept of Circle - Centre, Radius, Diameter, Arc, Sector, Chord, Segment, Semicircle, Circumference, Interior and Exterior, Concentric Circles
- Tangent to a Circle
- Number of Tangents from a Point on a Circle
- Concept of Circle - Centre, Radius, Diameter, Arc, Sector, Chord, Segment, Semicircle, Circumference, Interior and Exterior, Concentric Circles

##### Triangles

- Similar Figures
- Similarity of Triangles
- Basic Proportionality Theorem Or Thales Theorem
- Criteria for Similarity of Triangles
- Areas of Similar Triangles
- Right-angled Triangles and Pythagoras Property
- Similarity Triangle Theorem
- Application of Pythagoras Theorem in Acute Angle and Obtuse Angle
- Triangles Examples and Solutions
- Angle Bisector
- Similarity
- Ratio of Sides of Triangle

##### Constructions

##### Trigonometry

##### Heights and Distances

##### Trigonometric Identities

##### Introduction to Trigonometry

##### Statistics and Probability

##### Probability

##### Statistics

##### Coordinate Geometry

##### Lines (In Two-dimensions)

##### Mensuration

##### Areas Related to Circles

##### Surface Areas and Volumes

#### notes

You may recall that an equation is called an identity when it is true for all values of the variables involved. Similarly, an equation involving trigonometric ratios of an angle is called a trigonometric identity, if it is true for all values of the angle(s) involved.

In ∆ ABC, right-angled at B, we have;

`"AB"^2 + "BC"^2 = "AC"^2` (1)

Dividing each term of (1) by `"AC"^2`, we get

`"AB"^2/"AC"^2` +`"BC"^2/"AC"^2`= `"AC"^2/"AC"^2`

`("AB"/"AC")^2 + ("BC"/"AC")^2= ("AC"/"AC")^2`

`(cos A)^2 + (sin A)^2 = 1`

`cos^2 A + sin^2 A = 1` (2)

(i) `cos^2 A = 1- sin^2 A`

(ii) `sin^2 A = 1-cos^2 A`

This is true for all A such that 0° ≤ A ≤ 90°. So, this is a trigonometric identity.

Let us now divide (1) by `"AB"^2`. We get

`"AB"^2/"AB"^2 + "BC"^2/"AB"^2= "AC"^2/"AB"^2`

`("AB"/"AB")^2 + ("BC"/"AB")^2= ("AC"/"AB")^2`

`1+tan^2A= sec^2A` (3)

(i) `tan^2A= sec^2A-1`

(ii)`sec^2A- tan^2A= 1`

This is true for all A such that 0° ≤ A < 90°.

Let us see what we get on dividing (1) by `"BC"^2`. We get,

`"AB"^2/"BC"^2 +"BC"^2/"BC"^2= "AC"^2/"BC"^2`

`("AB"/"BC")^2 + ("BC"/"BC")^2= ("AC"/"BC")^2`

`cot^2A+1= cosec^2A` (4)

(i)`cot^2 A= cosec^2 A- 1`

(ii)`cosec^2 A- cot^2 A= 1`

Example 1 : Express the ratios cos A, tan A and sec A in terms of sin A.

Solution : Since `cos^2 A + sin^2 A = 1,` therefore,

`cos^2 A = 1 – sin^2 A`, i.e., `cos A = +or- sqrt(1-sin^2A)`

This gives `cos A = sqrt(1-sin^2A)`

Hence, `tanA= "sinA"/"cosA"= "sinA"/sqrt(1-sin^2A)` and

`sec A= 1/"cosA"= 1/sqrt(1-sin^2A)`

#### Video Tutorials

#### Shaalaa.com | Proving Identities (example 5a)

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