7.1Simplifying and Proving Trigonometric Identities
Apply Pythagorean identities (sin²θ + cos²θ = 1 and variants), reciprocal identities, and quotient identities to simplify trig expressions and prove identities.
Trig identities are the algebraic tools for simplifying and solving trig equations. The Pythagorean, reciprocal, and quotient identities are used in every calculus integration technique involving trig substitution.
How can you use known trigonometric relationships to rewrite an expression in a simpler or more useful form?
A trigonometric identity is an equation that is true for all values of the variable for which both sides are defined. Unlike a conditional equation (which may be true only for specific values), an identity holds universally.
The three families of fundamental identities are:
Pythagorean
sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = csc²θ
Reciprocal
cscθ = 1/sinθ
secθ = 1/cosθ
cotθ = 1/tanθ
Quotient
tanθ = sinθ/cosθ
cotθ = cosθ/sinθ
Strategy for proving identities: Work on one side only (usually the more complex side). Rewrite using known identities, factor, combine fractions, or multiply by a conjugate until the two sides match. Never move terms across the equals sign.
Simplify: sin²θ · sec²θ + sin²θ
Factor out sin²θ: sin²θ(sec²θ + 1)
Replace sec²θ + 1 using the Pythagorean identity 1 + tan²θ = sec²θ → sec²θ + 1 = tan²θ + 2. Wait — let's try a cleaner approach.
Factor: sin²θ · sec²θ + sin²θ = sin²θ(sec²θ + 1)
Use sec²θ = 1/cos²θ: sin²θ/cos²θ + sin²θ = tan²θ + sin²θ
Alternatively, factor differently: sin²θ(1/cos²θ + 1) = sin²θ · (1 + cos²θ)/cos²θ = tan²θ · (1 + cos²θ) — this doesn't simplify further, so the simplest form is tan²θ + sin²θ.
Simplify: (1 − cos²θ)/sinθ
Recognize 1 − cos²θ = sin²θ (Pythagorean identity).
Substitute: sin²θ / sinθ
Cancel one factor of sinθ (assuming sinθ ≠ 0): sinθ
Prove the identity: tanθ · cosθ = sinθ
Start with the left side: tanθ · cosθ
Replace tanθ with sinθ/cosθ (quotient identity): (sinθ/cosθ) · cosθ
Cancel cosθ: sinθ
Left side equals right side. ✓
Prove: secθ − cosθ = sinθ · tanθ
Work on the left side: secθ − cosθ
Replace secθ = 1/cosθ: 1/cosθ − cosθ
Combine over a common denominator: (1 − cos²θ)/cosθ
Use 1 − cos²θ = sin²θ: sin²θ/cosθ
Split: sinθ · (sinθ/cosθ) = sinθ · tanθ ✓
Simplify: (sec²θ − 1)/sec²θ
Use sec²θ − 1 = tan²θ (Pythagorean identity): tan²θ/sec²θ
Replace sec²θ = 1/cos²θ: tan²θ · cos²θ
Replace tan²θ = sin²θ/cos²θ: (sin²θ/cos²θ) · cos²θ
Cancel cos²θ: sin²θ
Simplify: cos²θ · csc²θ − cos²θ
Hint: Factor out cos²θ, then use a Pythagorean identity on what remains inside the parentheses.
Simplify: sinθ · cotθ
Hint: Replace cotθ with cosθ/sinθ and cancel.
Prove: cscθ · cosθ = cotθ
Hint: Replace cscθ with 1/sinθ on the left side.
Prove: (1 + tanθ)² = sec²θ + 2tanθ
Hint: Expand the left side, then use 1 + tan²θ = sec²θ.
Simplify: (csc²θ − 1)/csc²θ
Hint: Use csc²θ − 1 = cot²θ, then rewrite cot²θ/csc²θ in terms of sin and cos.
Interactive Practice — 5 Questions
Which identity states that sin²θ + cos²θ = 1?
Simplify: sinθ/cosθ
Which expression is equivalent to 1 − sin²θ?
Simplify: cosθ · tanθ
To prove an identity, you should:
Common Mistakes
Moving terms across the equals sign when proving an identity (e.g., adding cosθ to both sides).
Work on one side only. Transform it using identities until it equals the other side.
Confusing sec²θ − 1 = tan²θ with sec²θ + 1 = tan²θ.
The correct form is sec²θ − 1 = tan²θ (subtract 1 from sec²θ = 1 + tan²θ).
Writing 1/sin²θ = cos²θ.
1/sin²θ = csc²θ. The reciprocal of sin²θ is csc²θ, not cos²θ.
Cancelling sinθ from sinθ + cos²θ/sinθ as if sinθ were a factor of the whole expression.
Only cancel common factors. Here sinθ is a term in the numerator, not a factor of the entire numerator.
Math Tips
When stuck proving an identity, try converting everything to sinθ and cosθ — it almost always reveals the path.
The three Pythagorean identities all come from one equation: sin²θ + cos²θ = 1. Divide both sides by cos²θ to get 1 + tan²θ = sec²θ; divide by sin²θ to get cot²θ + 1 = csc²θ.
If you see a difference of squares like 1 − cos²θ or sec²θ − 1, immediately replace it with the Pythagorean equivalent.
Multiplying numerator and denominator by a conjugate (e.g., 1 + sinθ) is a powerful technique when you see expressions like 1 − sinθ in a denominator.
Always check the domain: identities hold wherever both sides are defined. Note that tanθ is undefined at θ = π/2 + nπ.