Verifying Trig Identities
A free Trigonometry lesson from the “Trig Identities” unit, with a worked example and practice problems including step-by-step solutions.
Verifying an identity means showing two expressions are equal for every allowed angle, not solving for a value. You work on ONE side at a time and transform it with known identities (tan = sin/cos, sin^2 + cos^2 = 1, reciprocals) until it matches the other side; you may never move terms across the equals sign as if it were an equation. A good first step usually rewrites everything in sines and cosines, applies a Pythagorean identity, or combines fractions over a common denominator.
What you'll learn
- State what a verification requires and how it differs from solving an equation
- Choose a sound first step (convert to sin/cos, use a Pythagorean identity, combine fractions)
- Carry out short multi-step verifications that transform one side into the other
Worked example
Problem. Verify the identity tan(theta)cos(theta) = sin(theta). Show the steps that turn the left side into the right side.
- Start with the left side and rewrite tan(theta) as sin(theta)/cos(theta).
- This gives (sin(theta)/cos(theta))*cos(theta); cancel the common factor cos(theta).
- The left side becomes sin(theta), which equals the right side, so the identity is verified.
Answer: tan(theta)cos(theta) = (sin(theta)/cos(theta))*cos(theta) = sin(theta)
Practice problems
1. To verify an identity, what must you show?
Choices: The two sides are equal for every angle where both are defined · The equation is true for one chosen angle · A value of theta that makes the equation true · The two sides are approximately equal
Show solution
- A verification is not solving for one value of theta.
- You must show the sides are identical wherever they are defined.
- So the correct goal is equality for all allowed angles.
Answer: The two sides are equal for every angle where both are defined
2. Which is a sound first step for verifying tan(theta)cot(theta) = 1?
Choices: Rewrite tan and cot in terms of sin and cos · Cross-multiply across the equals sign · Add 1 to both sides · Square both sides
Show solution
- You may not move terms across the equals sign in a verification.
- Converting to sin and cos exposes the cancellation.
- tan*cot = (sin/cos)(cos/sin) = 1.
Answer: Rewrite tan and cot in terms of sin and cos
3. Fill the missing right side after the first step: cot(theta)sin(theta) = (cos(theta)/sin(theta))*sin(theta) = ?
Show solution
- Rewrite cot(theta) as cos(theta)/sin(theta).
- Multiply by sin(theta) and cancel the sin(theta) factor.
- The result is cos(theta).
Answer: cos(theta)
4. Verify sec(theta)cos(theta) = 1. State the single expression the left side becomes after rewriting sec(theta) and canceling.
Show solution
- Rewrite sec(theta) as 1/cos(theta).
- Then (1/cos(theta))*cos(theta) cancels the cosine.
- The left side equals 1, matching the right side.
Answer: 1
5. Verify (1 - cos^2(theta))/sin(theta) = sin(theta). After using a Pythagorean identity, the numerator 1 - cos^2(theta) becomes what expression?
Show solution
- From sin^2(theta) + cos^2(theta) = 1, subtract cos^2(theta).
- So 1 - cos^2(theta) = sin^2(theta).
- Then sin^2(theta)/sin(theta) = sin(theta), verifying the identity.
Answer: sin^2(theta)
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