# Proofs of trigonometric identities

226,661pages on
this wiki

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

$\tan \theta = \frac {\sin \theta}{\cos \theta}$

Proof:

1. $\frac{\sin \theta}{\cos \theta} = \frac{\frac{opposite}{hypotenuse}}{\frac{adjacent}{hypotenuse}}$ (definition of the sine and cosine)
2. $\frac{\sin \theta}{\cos \theta} = \frac{\frac{opposite}{hypotenuse} \frac{hypotenuse}1}{\frac{adjacent}{hypotenuse} \frac{hypotenuse}1}$ (multiply numerator and denominator by hypotenuse)
3. $\frac{\sin \theta}{\cos \theta} = \frac{opposite}{adjacent} = \tan \theta$ (definition of tangent)

$\tan \theta = \frac {\sec \theta}{\csc \theta}$

Proof:

1. $\frac {\sec \theta}{\csc \theta} = \frac{\frac{hypotenuse}{adjacent}}{\frac{hypotenuse}{opposite}}$ (definition of secant and cosecant)
2. $\frac{\sec \theta}{\csc \theta} = \frac{\frac{hypotenuse}{adjacent} \frac{1}{hypotenuse}}{\frac{hypotenuse}{opposite}\frac{1}{hypotenuse}}$ (multiply numerator and denominator by $\frac{1}{hypotenuse}$)
3. $\frac {\sec \theta}{\csc \theta} = \frac{\frac{1}{adjacent}}{\frac{1}{opposite}}$ (Simplify)
4. $\frac {\sec \theta}{\csc \theta} = \frac{opposite}{adjacent} = \tan \theta$ (definition of the tangent)

#### Cotangent

$\cot \theta = \frac {\cos \theta}{\sin \theta}$

Proof:

1. $\frac{\cos \theta}{\sin \theta} = \frac{\frac{adjecent}{hypotenuse}}{\frac{opposite}{hypotenuse}}$ (definition of the sine and cosine)
2. $\frac{\cos \theta}{\sin \theta} = \frac{\frac{adjacent}{hypotenuse} \frac{hypotenuse}1}{\frac{opposite}{hypotenuse} \frac{hypotenuse}1}$ (multiply numerator and denominator by hypotenuse)
3. $\frac{\cos \theta}{\sin \theta} = \frac{adjacent}{opposite} = \cot \theta$ (definition of cotangent)

$\cot \theta = \frac {\csc \theta}{\sec \theta}$

Proof:

1. $\frac {\csc \theta}{\sec \theta} = \frac{\frac{hypotenuse}{opposite}}{\frac{hypotenuse}{adjacent}}$ (definition of secant and cosecant)
2. $\frac{\csc \theta}{\sec \theta} = \frac{\frac{hypotenuse}{opposite} \frac{1}{hypotenuse}}{\frac{hypotenuse}{adjacent}\frac{1}{hypotenuse}}$ (multiply numerator and denominator by $\frac{1}{hypotenuse}$)
3. $\frac {\sec \theta}{\csc \theta} = \frac{\frac{1}{opposite}}{\frac{1}{adjacent}}$ (Simplify)
4. $\frac {\sec \theta}{\csc \theta} = \frac{adjacent}{opposite} = \cot \theta$ (definition of the cotangent)

### Inequalities

#### Sine

$\frac{\sin \theta}{\theta} < 1$

Proof: Template:Sectstub

#### Tangent

$\frac{\tan \theta}{\theta} > 1$

Proof: Template:Sectstub

### Pythagorean identities

#### Basic Pythagorean identity

$\sin^2(x) + \cos^2(x) = 1 \!$

Proof:

1. $a^2 + b^2 = h^2 \!\$ (Pythagorean Theorem)
2. $a = h \sin \theta \!\$ and $b = h \cos \theta \!\$ (definition of sine and cosine)
3. $a^2 = h^2 \sin^2 \theta \!\$ and $b^2 = h^2 \cos^2 \theta \!\$ (square both sides of each equation)
4. $h^2 \sin^2(x) + h^2 \cos^2(x) = h^2 \!\$ (substitute expressions for a² and b² into first equation)
5. $\sin^2(x) + \cos^2(x) = 1 \!$ (Divide both sides by h²)

#### Tangent and Secant

$\tan^2 \theta + 1 = \sec^2 \theta \!\$

Proof:

1. $\sin^2 \theta + \cos^2 \theta = 1 \!\$ (Basic Pythagorean identity)
2. $\frac{\sin^2 \theta}{\cos^2 \theta} + \frac{\cos^2 \theta}{\cos^2 \theta} = \frac 1{\cos^2 \theta}$ (divide both sides by $\cos^2 \theta \$)
3. $\tan^2 \theta + 1 = \sec^2 \theta \!\$ (tangent ratio and definition of secant)

#### Cotangent and Cosecant

$\cot^2 \theta + 1 = \csc^2 \theta \!\$

Proof:

1. $\cos^2 \theta + \sin^2 \theta = 1 \!\$ (Basic Pythagorean identity)
2. $\frac{\cos^2 \theta}{\sin^2 \theta} + \frac{\sin^2 \theta}{\sin^2 \theta} = \frac 1{\sin^2 \theta}$ (divide both sides by $\sin^2 \theta \$)
3. $\cot^2 \theta + 1 = \csc^2 \theta \!\$ (cotangent ratio and definition of cosecant)

## Identities involving Calculus

### Sine and angle ratio identity

$\lim_{\theta \to 0}{\frac{\sin \theta}{\theta}} = 1$

Proof:

1. $\frac{\tan \theta}{\theta} > 1$ (tangent inequality identity)
2. $\frac{\tan \theta}{\theta} = \frac{\sin \theta}{\theta \cos \theta} > 1$ (tanget ratio identity)
3. $\frac{\sin \theta}{\theta} > \cos \theta$ (multiply both sides by $\cos \theta \!\$)
4. $1 > \frac{\sin \theta}{\theta} > \cos \theta$ (sine inequality identity)
5. $\lim_{\theta \to 0} 1 = 1$ and $\lim_{\theta \to 0} \cos \theta = 1$ ($\cos 0 = 0 \$)
6. $\lim_{\theta \to 0}{\frac{\sin \theta}{\theta}} = 1$ (Squeeze Theorem)

### Cosine and angle ratio identity

$\lim_{\theta \to 0}\frac{\cos \theta - 1}{\theta} = 0$

Proof:

1. $\lim_{\theta \to 0}\frac{\cos \theta - 1}{\theta} = \lim_{\theta \to 0}\frac{\cos^2 \theta - 1}{\theta (\cos \theta + 1)}$ (multiply numerator and denominator by $\cos \theta + 1 \$)
2. $\cos^2(x) - 1 = -\sin^2(x)\$ (Basic Pythagorean identity)
3. $\lim_{\theta \to 0}\frac{\cos \theta - 1}{\theta} = \lim_{\theta \to 0}\frac{-\sin^2 \theta}{\theta(\cos \theta + 1)}$ (substitution)
4. $\lim_{\theta \to 0}\frac{\cos \theta - 1}{\theta} = -\lim_{\theta \to 0} \frac{\sin \theta}{\theta} \lim_{\theta \to 0} \frac{\sin \theta}{\cos \theta + 1}$ (multiplication of limits property)
5. $-\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = -1$ (sine and angle ratio identity)
6. $\lim_{\theta \to 0} \frac{\sin \theta}{\cos \theta + 1} = \frac{0}{2} = 0$ (since $\sin 0 = 0 \$ and $\cos 0 = 1 \$)
7. $\lim_{\theta \to 0}\frac{\cos \theta - 1}{\theta}= (-1)(0) = 0 \$ (from steps 5 and 6)

### Cosine and square of angle ratio identity

$\lim_{\theta \to 0}\frac{1 - \cos \theta}{\theta^2} = \frac{1}{2}$

Proof:

1. $(1 - \cos \theta)(1 + \cos \theta) = 1 - \cos^2 \theta \$ (polynomial multiplication)
2. $\lim_{\theta \to 0}\frac{1 - \cos \theta}{\theta^2} = \lim_{\theta \to 0} \ \frac{1 - \cos^2 \theta}{\theta^2 \ (1 + \cos \theta)}$ (multiply both sides by $(1 + \cos \theta)\,$)
3. $1 - \cos^2 \theta = \sin^2 \theta \$ (basic Pythagorean identity)
4. $\lim_{\theta \to 0}\frac{1 - \cos \theta}{\theta^2}=\lim_{\theta \to 0} \ \frac{\sin^2 \theta}{\theta^2 \ (1 + \cos \theta)}$ (substitution)
5. $\lim_{\theta \to 0}\frac{1 - \cos \theta}{\theta^2}=\bigg(\lim_{\theta \to 0}\frac{\sin \theta}{\theta}\bigg)^2 \ \lim_{\theta \to 0}\frac{1}{1 + \cos \theta}$ (limit properties)
6. $\bigg( \lim_{\theta \to 0}\frac{\sin \theta}{\theta} \bigg)^2 = 1$ (sine and angle ratio identity)
7. $\lim_{\theta \to 0}\frac{1}{1 + \cos \theta} = \frac 1 2$ (since $\cos 0 = 1 \$)
8. $\lim_{\theta \to 0}\frac{1 - \cos \theta}{\theta^2} = \frac{1}{2}$ (steps 6 and 7)

### Identity 4: $\lim_{\phi \to 0}\frac{\sin k\phi}{p\phi} = \frac{k}{p}$

#### Proof

1. $= \lim_{\phi \to 0}\frac{\sin k\phi}{p\phi} \frac{k}{k}$ (multiplicative identity)
2. $= \lim_{\phi \to 0}\frac{k}{p}\frac{\sin k\phi}{k\phi}$ (reorder variables in denominator)
3. $= \frac{k}{p} \lim_{\phi \to 0}\frac{\sin k\phi}{k\phi}$ (limit laws)
4. $\lim_{\phi \to 0}\frac{\sin k\phi}{k\phi} = \lim_{k\phi \to 0}\frac{\sin k\phi}{k\phi}$ Substituting θ for kφ, we get Identity 1. Thus,
5. $= \frac{k}{p} (1) = \frac{k}{p}$