The Pythagorean Identity

The most fundamental trigonometric identity comes from the unit circle equation x² + y² = 1. Since x = cos θ and y = sin θ:

sin² θ + cos² θ = 1

This identity can be rearranged to give two useful forms:

sin² θ = 1 − cos² θ

cos² θ = 1 − sin² θ

Quotient and Reciprocal Identities

These identities express relationships between the trigonometric functions:

Quotient Identities

tan θ = sin θ/cos θ

cot θ = cos θ/sin θ

Reciprocal Identities

cosec θ = 1/sin θ

sec θ = 1/cos θ

cot θ = 1/tan θ

How to Prove Trigonometric Identities

To prove an identity, you show that the left-hand side (LHS) is equal to the right-hand side (RHS). Key strategies include:

1

Work with one side only: Choose the more complex side and simplify it to equal the other side.

2

Convert everything to sin and cos: Replace tan, sec, cosec and cot with their definitions in terms of sin and cos.

3

Use the Pythagorean identity: Replace sin²θ with 1 − cos²θ (or vice versa) when needed.

4

Factorise or expand: Look for common factors and difference-of-squares patterns.

Key Vocabulary

Identity

An equation that is true for all valid values of the variable, not just specific ones.

Pythagorean Identity

sin²θ + cos²θ = 1, derived from the unit circle and Pythagoras' theorem.

LHS / RHS

Left-Hand Side and Right-Hand Side of an equation. In proofs, we show LHS = RHS.

Reciprocal

The multiplicative inverse of a number. For example, the reciprocal of sin θ is cosec θ.

Worked Examples

1

Simplify: sin² θ + cos² θ + tan² θ

Step 1: Use the Pythagorean identity: sin²θ + cos²θ = 1.

Step 2: So the expression becomes 1 + tan²θ.

Answer: 1 + tan² θ (which also equals sec² θ).

2

Prove that sin θ/cos θ + cos θ/sin θ = 1/sin θ cos θ

LHS: sin θ/cos θ + cos θ/sin θ

Step 1: Common denominator = sin θ cos θ:

= sin² θ + cos² θ/sin θ cos θ

Step 2: Apply Pythagorean identity: sin²θ + cos²θ = 1.

Result: = 1/sin θ cos θ = RHS. Identity proven.

3

Show that (1 − cos² θ)(1 + cot² θ) = 1

Step 1: 1 − cos²θ = sin²θ (Pythagorean identity).

Step 2: cot²θ = cos²θ/sin²θ, so 1 + cot²θ = 1 + cos²θ/sin²θ = (sin²θ + cos²θ)/sin²θ = 1/sin²θ.

Step 3: LHS = sin²θ × 1/sin²θ = 1 = RHS.

Knowledge Check

Select the correct answer for each question. Click "Check Answer" to see if you are right.

Question 1

What does sin² θ + cos² θ equal?

1 0 2 tan² θ

Question 2

Which of the following equals tan θ?

cos θ/sin θ sin θ/cos θ 1/cos θ sin θ × cos θ

Question 3

Simplify: 1 − sin² θ

tan² θ sin² θ cos² θ 1

Question 4

What is sec θ equal to?

1/sin θ 1/tan θ sin θ + cos θ 1/cos θ

Question 5

Simplify: sin² θ/1 − cos² θ

1 0 tan² θ cos² θ

Trigonometric Identities