# Trigonometric Identities for Class 10

Class 10Trigonometric identities for class 10 are fundamental relationships that exist between the ratios of the sides of a right-angled triangle. These identities are extensively used in mathematics, science, and engineering to solve complex problems related to angles and distances. In this article, we will explore the various trigonometric identities and how they can be derived and applied in different contexts. We will also discuss the properties and applications of sine, cosine, tangent, cotangent, secant, and cosecant functions. Whether you are a student or a professional, understanding trigonometric identities is essential for solving a wide range of problems in various fields. So, let's delve into the world of trigonometry and learn about these fascinating identities.

## Trignometric identities Definition

Trigonometric identities are a set of fundamental relationships that express the ratios between the sides of a right-angled triangle. These identities play a crucial role in mathematics, science, and engineering as they help solve complex problems related to angles and distances. In essence, trigonometric identities are expressions that are always true, regardless of the specific values of the angles or sides of the triangle. By understanding these identities, we can simplify complex trigonometric expressions, manipulate equations, and solve a wide range of problems in various fields.

### Lists of trignometric identities

- Pythagorean Identity: sin²θ + cos²θ = 1
- Reciprocal Identity: cscθ = 1/sinθ, secθ = 1/cosθ, cotθ = 1/tanθ
- Quotient Identity: tanθ = sinθ/cosθ, cotθ = cosθ/sinθ
- Co-function Identity: sin(π/2 - θ) = cosθ, cos(π/2 - θ) = sinθ, tan(π/2 - θ) = cotθ
- Even-Odd Identities: sin(-θ) = -sinθ, cos(-θ) = cosθ, tan(-θ) = -tanθ
- Sum and Difference Identities: sin(A ± B) = sinAcosB ± cosAsinB, cos(A ± B) = cosAcosB ∓ sinAsinB, tan(A ± B) = (tanA ± tanB)/(1 ∓ tanAtanB)
- Double Angle Identities: sin2θ = 2sinθcosθ, cos2θ = cos²θ - sin²θ, tan2θ = (2tanθ)/(1 - tan²θ)
- Half-Angle Identities: sin(θ/2) = ±√[(1 - cosθ)/2], cos(θ/2) = ±√[(1 + cosθ)/2], tan(θ/2) = ±√[(1 - cosθ)/(1 + cosθ)]
- Product-to-Sum Identities: sinA sinB = (cos(A - B) - cos(A + B))/2, cosA cosB = (cos(A - B) + cos(A + B))/2, sinA cosB = (sin(A + B) + sin(A - B))/2, cosA sinB = (sin(A + B) - sin(A - B))/2

These identities can be used to simplify complex trigonometric expressions, prove mathematical theorems, and solve problems in various fields, such as physics, engineering, and surveying.

### Reciprocal Trigonometric Identities

The reciprocal trigonometric identities are:

- Sin θ = 1/Csc θ or Csc θ = 1/Sin θ
- Cos θ = 1/Sec θ or Sec θ = 1/Cos θ
- Tan θ = 1/Cot θ or Cot θ = 1/Tan θ

### Pythagorean Trigonometric Identities

There are three Pythagorean trigonometric identities in trigonometry that are based on the right-triangle theorem or Pythagoras theorem.

- sin
^{2 }a + cos^{2 }a = 1 - 1+tan
^{2}a = sec^{2}a - cosec
^{2 }a = 1 + cot^{2 }a

### Ratio Trigonometric Identities

The trigonometric ratio identities are:

- Tan θ = Sin θ/Cos θ
- Cot θ = Cos θ/Sin θ

### Trigonometric Identities of Opposite Angles

The list of opposite angle trigonometric identities are:

- Sin (-θ) = – Sin θ
- Cos (-θ) = Cos θ
- Tan (-θ) = – Tan θ
- Cot (-θ) = – Cot θ
- Sec (-θ) = Sec θ
- Csc (-θ) = -Csc θ

### Trigonometric Identities of Complementary Angles

In geometry, two angles are complementary if their sum is equal to 90 degrees. Similarly, when we can learn here the trigonometric identities for complementary angles.

- Sin (90 – θ) = Cos θ
- Cos (90 – θ) = Sin θ
- Tan (90 – θ) = Cot θ
- Cot ( 90 – θ) = Tan θ
- Sec (90 – θ) = Csc θ
- Csc (90 – θ) = Sec θ

### Trigonometric Identities of Supplementary Angles

Two angles are supplementary if their sum is equal to 90 degrees. Similarly, when we can learn here the trigonometric identities for supplementary angles.

- sin (180°- θ) = sinθ
- cos (180°- θ) = -cos θ
- cosec (180°- θ) = cosec θ
- sec (180°- θ)= -sec θ
- tan (180°- θ) = -tan θ
- cot (180°- θ) = -cot θ

### Sum and Difference of Angles Trigonometric Identities

Consider two angles , A and B, the trigonometric sum and difference identities are as follows:

Sine: sin(A ± B) = sinAcosB ± cosAsinB

Cosine: cos(A ± B) = cosAcosB ∓ sinAsinB

Tangent: tan(A ± B) = (tanA ± tanB)/(1 ∓ tanAtanB)

These identities can be proven using the angle addition formulas and the Pythagorean identity. The angle addition formulas express sin(A ± B) and cos(A ± B) in terms of sinA, sinB, cosA, and cosB. The Pythagorean identity is used to simplify the expressions involving sine and cosine.

### Double Angle Trigonometric Identities

If the angles are doubled, then the trigonometric identities for sin, cos and tan are:

- sin 2θ = 2 sinθ cosθ
- cos 2θ = cos
^{2}θ – sin^{2 }θ = 2 cos^{2}θ – 1 = 1 – 2sin^{2}θ - tan 2θ = (2tanθ)/(1 – tan
^{2}θ)

### Half Angle Identities

If the angles are halved, then the trigonometric identities for sin, cos and tan are:

- sin (θ/2) = ±√[(1 – cosθ)/2]
- cos (θ/2) = ±√(1 + cosθ)/2
- tan (θ/2) = ±√[(1 – cosθ)(1 + cosθ)]

### Product-Sum Trigonometric Identities

The product-sum trigonometric identities change the sum or difference of sines or cosines into a product of sines and cosines.

- Sin A + Sin B = 2 Sin(A+B)/2 . Cos(A-B)/2
- Cos A + Cos B = 2 Cos(A+B)/2 . Cos(A-B)/2
- Sin A – Sin B = 2 Cos(A+B)/2 . Sin(A-B)/2
- Cos A – Cos B = -2 Sin(A+B)/2 . Sin(A-B)/2

### Trigonometric Identities of Products

These identities are:

- Sin A. Sin B = [Cos (A – B) – Cos (A + B)]/2
- Sin A. Cos B = [Sin (A + B) + Sin (A – B)]/2
- Cos A. Cos B = [Cos (A + B) + Cos (A – B)]/2

## Trigonometric Identities Proofs

There are many trigonometric identities that can be proved using various algebraic and geometric manipulations. Here are a few examples:

**Pythagorean Identity:**

sin^2(x) + cos^2(x) = 1

Proof:

Consider a right-angled triangle with hypotenuse of length 1, and let the opposite and adjacent sides be sin(x) and cos(x) respectively. Then, by the Pythagorean Theorem, we have:

sin^2(x) + cos^2(x) = 1^2 = 1

Thus, the Pythagorean Identity is proven.

**Sum and Difference Identities:**

sin(a+b) = sin(a)cos(b) + cos(a)sin(b)

cos(a+b) = cos(a)cos(b) - sin(a)sin(b)

Proof:

We can prove the sum and difference identities using the angle addition formulas, which are:

sin(a+b) = sin(a)cos(b) + cos(a)sin(b)

cos(a+b) = cos(a)cos(b) - sin(a)sin(b)

To prove the first identity, we start with the left-hand side:

sin(a+b) = sin(a)cos(90-b) + cos(a)sin(90-b) (using complementary angles)

= sin(a)sin(b) + cos(a)cos(b) (using the sine and cosine of complementary angles)

= sin(a)cos(b) + cos(a)sin(b) (by rearranging terms)

To prove the second identity, we start with the left-hand side:

cos(a+b) = cos(a)cos(90-b) - sin(a)sin(90-b) (using complementary angles)

= cos(a)sin(b) - sin(a)cos(b) (using the sine and cosine of complementary angles)

= cos(a)cos(b) - sin(a)sin(b) (by rearranging terms)

Thus, both the sum and difference identities are proven.

**Double Angle Identities:**

sin(2x) = 2sin(x)cos(x)

cos(2x) = cos^2(x) - sin^2(x)

Proof:

To prove the double angle identities, we can use the sum and difference identities. First, we start with the first identity:

sin(2x) = sin(x+x) = sin(x)cos(x) + cos(x)sin(x) = 2sin(x)cos(x)

To prove the second identity, we start with the sum identity:

cos(2x) = cos(x+x) = cos(x)cos(x) - sin(x)sin(x) = cos^2(x) - sin^2(x)

Thus, both double angle identities are proven.

These are just a few examples of trigonometric identities that can be proven. There are many more, and the proofs can be quite complex depending on the identity being proven.

### Sine, cosine, Tangent rule

The Sine, Cosine, and Tangent rules (also known as trigonometric ratios or functions) are fundamental concepts in trigonometry, which is a branch of mathematics that deals with the relationships between the angles and sides of triangles. These rules can be used to solve a wide range of problems involving triangles, including finding unknown sides and angles.

**Sine rule:** The sine rule relates the lengths of the sides of a triangle to the sine of the angles opposite those sides. It states that in any triangle ABC, the ratio of the length of a side to the sine of the angle opposite that side is the same for all three sides:

a / sin(A) = b / sin(B) = c / sin(C)

where a, b, and c are the lengths of the sides, and A, B, and C are the angles opposite those sides, respectively.

**Cosine rule:** The cosine rule relates the lengths of the sides of a triangle to the cosine of one of its angles and the other two sides. It states that in any triangle ABC, the square of the length of a side is equal to the sum of the squares of the other two sides minus twice the product of those sides and the cosine of the angle between them:

a^2 = b^2 + c^2 - 2bc cos(A) b^2 = a^2 + c^2 - 2ac cos(B) c^2 = a^2 + b^2 - 2ab cos(C)

where a, b, and c are the lengths of the sides, and A, B, and C are the angles opposite those sides, respectively.

**Tangent rule**: The tangent rule relates the lengths of the sides of a right triangle to the tangent of one of its acute angles. It states that in any right triangle ABC with angle A being the acute angle, tan(A) = opposite / adjacent where opposite is the length of the side opposite to angle A and adjacent is the length of the side adjacent to angle A.

### Conclusion

Trigonometric identities are a fundamental concept in mathematics, particularly in trigonometry, which deals with the relationships between the angles and sides of triangles. These identities are used to solve a wide range of problems involving triangles, including finding unknown sides and angles.

For Class 10 students, it is important to understand the basic trigonometric identities, such as the Sine, Cosine, and Tangent rules. The Sine rule relates the lengths of the sides of a triangle to the sine of the angles opposite those sides, while the Cosine rule relates the lengths of the sides to the cosine of one of its angles and the other two sides. The Tangent rule, on the other hand, relates the lengths of the sides of a right triangle to the tangent of one of its acute angles.

These trigonometric identities are crucial for Class 10 students, as they provide the foundation for more advanced concepts in trigonometry. By mastering these identities, students can solve a variety of problems involving triangles and further their understanding of mathematics. Therefore, it is important to practice and apply these identities regularly in order to build a strong foundation in trigonometry.