Simplifying Trig Expressions
A free Trigonometry lesson from the “Trig Identities” unit, with a worked example and practice problems including step-by-step solutions.
Simplifying trig expressions means replacing pieces with equivalent forms and canceling only when algebra allows it. In this lesson, the goal is to simplify expressions by substituting identities carefully. Prerequisite check: Algebra II or College Algebra foundations. Example 1: sin^2(x) + cos^2(x) = 1 comes from the unit circle. Example 2: tan(x)cos(x) simplifies to sin(x) because tan(x) = sin(x)/cos(x). A common mistake is canceling across addition or treating an identity as something to solve for one angle; the safer habit is to rewrite unfamiliar pieces using reciprocal, quotient, or Pythagorean identities.
What you'll learn
- Simplify expressions by substituting identities carefully
- replace expressions only with equivalent trig relationships
- Explain why identities make complicated trig expressions readable
Worked example
Problem. Example 1 Foundation: What makes an equation a trig identity?
- An identity is a relationship, not a single-angle answer.
- It holds wherever both sides are defined.
- Domain restrictions still matter.
Answer: it is true for every allowed angle
Practice problems
1. Practice 1 Foundation: Which reciprocal identity is correct?
Choices: sec(x) = 1/cos(x) · sec(x) = 1/sin(x) · csc(x) = 1/cos(x) · cot(x) = 1/tan(x) + 1
Show solution
- Warm-up: First identify exactly what the question is asking: Practice 1 Foundation: Which reciprocal identity is correct?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Secant is the reciprocal of cosine.
- So sec(x) = 1/cos(x).
- Keep each reciprocal pair straight.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: sec(x) = 1/cos(x)
2. Practice 2 Setup: Which quotient identity is correct?
Choices: tan(x) = sin(x)/cos(x) · tan(x) = cos(x)/sin(x) · sin(x) = tan(x)/cos(x) · cos(x) = tan(x)/sin(x)
Show solution
- Warm-up: First identify exactly what the question is asking: Practice 2 Setup: Which quotient identity is correct?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Tangent compares sine to cosine.
- The quotient identity is tan(x) = sin(x)/cos(x).
- Cotangent is the reciprocal quotient.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: tan(x) = sin(x)/cos(x)
3. Practice 3 Meaning: Simplify 1 - sin^2(x).
Choices: cos^2(x) · sin^2(x) · tan^2(x) · 1
Show solution
- Core Practice: First identify exactly what the question is asking: Practice 3 Meaning: Simplify 1 - sin^2(x).
- For data questions, identify what each statistic measures before calculating so the result matches the question.
- Use sin^2(x) + cos^2(x) = 1.
- Subtract sin^2(x) from both sides.
- 1 - sin^2(x) = cos^2(x).
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: cos^2(x)
4. Practice 4 Method: Simplify tan(x)cos(x).
Choices: sin(x) · cos(x) · tan(x) · 1
Show solution
- Core Practice: First identify exactly what the question is asking: Practice 4 Method: Simplify tan(x)cos(x).
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Replace tan(x) with sin(x)/cos(x).
- Then (sin(x)/cos(x))cos(x) = sin(x).
- Cancel only common factors.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: sin(x)
5. Practice 5 Reasoning: When verifying a trig identity, what is a safe strategy?
Choices: rewrite one side until it matches the other · divide both sides by anything that appears · plug in one angle and stop · move terms as if solving for x only
Show solution
- Core Practice: First identify exactly what the question is asking: Practice 5 Reasoning: When verifying a trig identity, what is a safe strategy?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Verification is about equivalence.
- Work one side into the other using identities.
- One numerical check is not a proof.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: rewrite one side until it matches the other
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