## Definition of Trigonometric Identities

## Sum and Difference Formulas of Trigonometry

tan (A + B) = tanA + tanB/ 1-tanAtanB

tan (A + B) = tan A – tanB/1 + tanAtanB

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## Education

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In geometry, you might have come across different types of triangles. The right-angle triangle is also one among them. In trigonometry, we study the sides and angles of a triangle. We make use of right-angle triangles to find the basic ratios of trigonometry and these ratios are then used in deriving various trigonometric identities and formulas. Let us learn the trigonometric identities and formulas in detail now.

The Trigonometric Identities are the equations that apply to Right Angled Triangles. Various trigonometric identities are employed in trigonometry formulas to solve problems based on the sides and angles of right-angled triangles. For the given angles, these trigonometry formulas include the basic trigonometric ratios of sine, cosine, tangent, secant, cosecant, and cotangent. These all trigonometric ratios are defined using the sides of the right angle triangle which has an adjacent side, opposite side, and hypotenuse side.

sin(A + B) = sin(A) cos(B) + cos(A) sin(B)

sin(A – B) = sin(A) cos(B) – cos(A) sin(B)

cos(A + B) = cos(A) cos(B) – sin(A) sin(B)

cos(A – B) = cos(A) cos(B) + sin(A) sin(B)

tan (A + B) = tanA + tanB/ 1-tanAtanB

tan (A + B) = tan A – tanB/1 + tanAtanB

1. sin θ = 1/cosecθ or cosecθ = 1/sin θ

2. cos θ = 1/sec θ or sec θ = 1/cos θ

3. tan θ = 1/cot θ or cotθ = 1/tan θ

1. tan θ = sinθ/cosθ

2. cot θ = cosθ/sinθ

You can derive the Pythagorean trigonometric identities from the Pythagoras theorem. Applying Pythagoras theorem to the right-angled triangle ABC, where BAC = θ

BC2 + AC2 = BA2

BC2/BA2 + AC2/BA2 = BA2/BA2

1. sin2θ + cos2θ = 1

This is one of the Pythagorean identities. Similarly, we can derive two other Pythagorean trigonometric identities.

2. 1 + tan2θ = sec2θ

3. 1 + cot2θ = cosec2θ

1. Sin (-θ) = – Sin θ

2. Cos (-θ) = Cos θ

3. Tan (-θ) = – Tan θ

4. Cot (-θ) = – Cot θ

5. Sec (-θ) = Sec θ

6. Cosec (-θ) = – Cosec θ

In geometry, two angles are said to be complementary if their sum is equal to 90 degrees. The complementary angles of trigonometry are

1. sin (90 – θ) = cos θ

2. cos (90 – θ) = sin θ

3. tan (90 – θ) = cot θ

4. cot ( 90 – θ) = tan θ

5. sec (90 – θ) = cosec θ

6. cosec (90 – θ) = sec θ

Two angles are supplementary if their sum is equal to 1800. The supplementary angles of trigonometry are

1. sin (1800- θ) = sinθ

2. cos (1800- θ) = – cos θ

3. cosec (1800- θ) = cosec θ

4. sec (1800- θ)= – sec θ

5. tan (1800- θ) = – tan θ

6. cot (1800- θ) = – cot θ

When the angles are doubled, then the trigonometric identities for sin, cos, and tan are written as

1. sin 2θ = 2sinθ cosθ

2. cos 2θ = cos2θ – sin2θ = 2cos2θ – 1 = 1 – 2sin2θ

3. tan 2θ = 2tanθ/1 – tan 2θ

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

1. sin(θ/2) = ± √[(1 – cosθ)/2]

2. cos(θ/2) = ± √(1 + cosθ)/2

3. tan(θ/2) = ± √[(1 – cosθ)(1 + cosθ)]

In trigonometric identities, the sum or difference of sines or cosines becomes a product of sines and cosines.

1. Sin A + Sin B = 2sin(A + B)/2 . cos(A-B)/2

2. Cos A + Cos B = 2cos(A + B)/2 . cos(A-B)/2

3. Sin A – Sin B = 2cos(A + B)/2 . sin(A – B)/2

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

Trigonometric product identities are:

1. Sin A. Sin B = cos(A – B) – cos(A+B)/2

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

3. Cos A. Cos B = Cos(A + B) – Cos(A – B)/2

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