# Proofs of trigonometric identities

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**Proofs of trigonometric identities** are used to show relations between trigonometric functions. This article will list trigonometric identities and prove them.

## Elementary trigonometric identities

### Ratio identities

#### Tangent

Proof:

- (definition of the sine and cosine)
- (multiply numerator and denominator by hypotenuse)
- (definition of tangent)

Proof:

- (definition of secant and cosecant)
- (multiply numerator and denominator by )
- (Simplify)
- (definition of the tangent)

#### Cotangent

Proof:

- (definition of the sine and cosine)
- (multiply numerator and denominator by hypotenuse)
- (definition of cotangent)

Proof:

- (definition of secant and cosecant)
- (multiply numerator and denominator by )
- (Simplify)
- (definition of the cotangent)

### Inequalities

#### Sine

Proof: Template:Sectstub

#### Tangent

Proof: Template:Sectstub

### Pythagorean identities

#### Basic Pythagorean identity

Proof:

- (Pythagorean Theorem)
- and (definition of sine and cosine)
- and (square both sides of each equation)
- (substitute expressions for a² and b² into first equation)
- (Divide both sides by h²)

#### Tangent and Secant

Proof:

- (Basic Pythagorean identity)
- (divide both sides by )
- (tangent ratio and definition of secant)

#### Cotangent and Cosecant

Proof:

- (Basic Pythagorean identity)
- (divide both sides by )
- (cotangent ratio and definition of cosecant)

## Identities involving Calculus

### Sine and angle ratio identity

Proof:

- (tangent inequality identity)
- (tanget ratio identity)
- (multiply both sides by )
- (sine inequality identity)
- and ()
- (Squeeze Theorem)

### Cosine and angle ratio identity

Proof:

- (multiply numerator and denominator by )
- (Basic Pythagorean identity)
- (substitution)
- (multiplication of limits property)
- (sine and angle ratio identity)
- (since and )
- (from steps 5 and 6)

### Cosine and square of angle ratio identity

Proof:

- (polynomial multiplication)
- (multiply both sides by )
- (basic Pythagorean identity)
- (substitution)
- (limit properties)
- (sine and angle ratio identity)
- (since )
- (steps 6 and 7)

### Identity 4:

#### Proof

- (multiplicative identity)
- (reorder variables in denominator)
- (limit laws)
- Substituting θ for kφ, we get Identity 1. Thus,