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 a trig expression means rewriting it as one clean function by substituting the reciprocal and quotient definitions (tan = sin/cos, sec = 1/cos, csc = 1/sin, cot = cos/sin) and then canceling. The Pythagorean identities sin^2 + cos^2 = 1, 1 + tan^2 = sec^2, and 1 + cot^2 = csc^2 are the main tools for collapsing sums and differences. The goal is always the simplest possible single function.
What you'll learn
- Replace tan, cot, sec, and csc with their sine/cosine definitions
- Apply the Pythagorean identities to collapse expressions
- Simplify a trig expression down to a single function
Worked example
Problem. Simplify sin(x)*sec(x).
- Rewrite sec(x) as 1/cos(x).
- Then sin(x)*(1/cos(x)) = sin(x)/cos(x).
- sin(x)/cos(x) is the definition of tan(x).
Answer: tan(x)
Practice problems
1. Simplify tan(x)*cos(x).
Show solution
- Warm-up: First identify exactly what the question is asking: Simplify tan(x)*cos(x).
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Replace tan(x) with sin(x)/cos(x).
- Multiply: (sin(x)/cos(x))*cos(x).
- Cancel cos(x) to get sin(x).
- Check the result by substituting or estimating: the response should match sin(x) and make sense in the original problem.
Answer: sin(x)
2. Simplify cos(x)*csc(x).
Show solution
- Warm-up: First identify exactly what the question is asking: Simplify cos(x)*csc(x).
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Replace csc(x) with 1/sin(x).
- Multiply: cos(x)*(1/sin(x)) = cos(x)/sin(x).
- cos(x)/sin(x) is the definition of cot(x).
- Check the result by substituting or estimating: the response should match cot(x) and make sense in the original problem.
Answer: cot(x)
3. Simplify sin(x)*sec(x).
Show solution
- Warm-up: First identify exactly what the question is asking: Simplify sin(x)*sec(x).
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Replace sec(x) with 1/cos(x).
- Multiply: sin(x)*(1/cos(x)) = sin(x)/cos(x).
- sin(x)/cos(x) is the definition of tan(x).
- Check the result by substituting or estimating: the response should match tan(x) and make sense in the original problem.
Answer: tan(x)
4. Simplify (1 - cos^2(x))/sin(x).
Show solution
- Core Practice: First identify exactly what the question is asking: Simplify (1 - cos^2(x))/sin(x).
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Use 1 - cos^2(x) = sin^2(x).
- The expression becomes sin^2(x)/sin(x).
- Cancel one sin(x) to get sin(x).
- Check the result by substituting or estimating: the response should match sin(x) and make sense in the original problem.
Answer: sin(x)
5. Simplify sec^2(x) - tan^2(x).
Show solution
- Core Practice: First identify exactly what the question is asking: Simplify sec^2(x) - tan^2(x).
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Use the identity 1 + tan^2(x) = sec^2(x).
- Rearrange to sec^2(x) - tan^2(x) = 1.
- The result is the constant 1.
- Check the result by substituting or estimating: the response should match 1 and make sense in the original problem.
Answer: 1
Practice this interactively with instant feedback and an AI tutor.
Practice Simplifying Trig Expressions Take the free placement check