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Equations Using Identities

A free Trigonometry lesson from the “Inverse Trig and Equations” unit, with a worked example and practice problems including step-by-step solutions.

Some trig equations mix different functions or arguments (like cos(2x) next to cos(x)), so you cannot solve them directly. The strategy is to first apply an identity — a double-angle formula such as cos(2x) = 2cos^2(x) - 1 or a Pythagorean identity — to rewrite everything in terms of a single trig function of x. Then the equation becomes a familiar factorable or quadratic form, and you solve each piece on [0, 2pi).

What you'll learn

Why it matters: Wave problems in physics and engineering — combining a signal and its frequency-doubled harmonic, or finding where overlapping oscillations cross — produce exactly these mixed-argument equations, and an identity is what untangles them.

Worked example

Problem. Solve cos(2x) = cos(x) on [0, 2pi).

  1. Replace cos(2x) with the identity 2cos^2(x) - 1, giving 2cos^2(x) - 1 = cos(x)
  2. Move all terms to one side and factor: 2cos^2(x) - cos(x) - 1 = 0 -> (2cos(x) + 1)(cos(x) - 1) = 0
  3. Solve cos(x) = 1 -> x = 0, and cos(x) = -1/2 -> x = 2pi/3, 4pi/3

Answer: x = 0, 2pi/3, 4pi/3

Practice problems

1. To solve cos(2x) = cos(x), which identity for cos(2x) is the most useful first step?

Choices: 2cos^2(x) - 1 · 1 - 2sin^2(x) · cos^2(x) - sin^2(x) · 2sin(x)cos(x)

Show solution
  1. Warm-up: First identify exactly what the question is asking: To solve cos(2x) = cos(x), which identity for cos(2x) is the most useful first step?
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. The equation already contains cos(x), so rewrite cos(2x) in terms of cosine only.
  4. 2cos^2(x) - 1 produces a quadratic purely in cos(x).
  5. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 2cos^2(x) - 1

2. Solve 2cos^2(x) - 1 = 0 on [0, 2pi). List all solutions.

Show solution
  1. Solve for cos: cos^2(x) = 1/2, so cos(x) = +/- sqrt(2)/2.
  2. The reference angle is pi/4, and cosine is +/- sqrt(2)/2 in all four quadrants.
  3. x = pi/4, 3pi/4, 5pi/4, 7pi/4.

Answer: x = pi/4, 3pi/4, 5pi/4, 7pi/4

3. Solve 2sin^2(x) - 1 = 0 on [0, 2pi). List all solutions.

Show solution
  1. sin^2(x) = 1/2, so sin(x) = +/- sqrt(2)/2.
  2. Reference angle pi/4; sine is +/- sqrt(2)/2 in all four quadrants.
  3. x = pi/4, 3pi/4, 5pi/4, 7pi/4.

Answer: x = pi/4, 3pi/4, 5pi/4, 7pi/4

4. Solve 2sin^2(x) = sin(x) on [0, 2pi). List all solutions.

Show solution
  1. Move all terms to one side: 2sin^2(x) - sin(x) = 0, then factor sin(x)(2sin(x) - 1) = 0.
  2. sin(x) = 0 -> x = 0, pi.
  3. sin(x) = 1/2 -> x = pi/6, 5pi/6.

Answer: x = 0, pi/6, 5pi/6, pi

5. Solve sin(2x) = cos(x) on [0, 2pi). List all solutions.

Show solution
  1. Use sin(2x) = 2sin(x)cos(x): 2sin(x)cos(x) - cos(x) = 0, then factor cos(x)(2sin(x) - 1) = 0.
  2. cos(x) = 0 -> x = pi/2, 3pi/2.
  3. sin(x) = 1/2 -> x = pi/6, 5pi/6.

Answer: x = pi/6, pi/2, 5pi/6, 3pi/2

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