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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.

Definition of Trigonometric Identities

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.

Sum and Difference Formulas of Trigonometry

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

Reciprocal Identities of Trigonometry

Reciprocal Identities of Trigonometry are

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

Trigonometric Ratio Identities

The trigonometric ratio Formulas are:

1. tan θ = sinθ/cosθ
2. cot θ = cosθ/sinθ

Pythagorean Identities of Trigonometry

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

Dividing Both Sides By Hypotenuse2

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

Trigonometric Identities of Opposite Angles

The Opposite Angle Trigonometric Formulas are:

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

Trigonometric Identities of Complementary Angles

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

Trigonometric Identities of Supplementary Angles

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

Double Angle Trigonometric Identities

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

Half Angle Identities

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)]

Product-Sum Trigonometric Identities

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 Formulas of Products

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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