# Trigonometric Identities PDF

### Trigonometric Identities

In mathematics, trigonometric identities are identities that involve trigonometric functions and are true for every single value of the variable in which they occur.

Geometrically, these are identities associated with certain functions of one or more angles. They differ from triangle identities, which involve both angles and sides.

length of a triangle. Only the former is covered in this article.

These identities are useful whenever expressions involving trigonometric functions need to be simplified.

An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function and then simplifying the result.

integral with trigonometric identities

### Basic trigonometric identities fall into four groups:

• Reciprocals
• Ratios
• Products
• Pythagorean Theorem

### 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.

• sina + cosa = 1
• 1+tan2 a  = sec2 a
• coseca = 1 + cota

### 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

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