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Mathematics, five units · Ages 17–18

Proving Trigonometric Identities Practice Sheet

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For each problem, choose one side to start with and write out the algebraic transformations until it equals the target side. Do not move terms across the equals sign.

  1. Prove that tan(x) * cos(x) = sin(x).

  2. Prove that 1 - sin^2(x) = cos^2(x).

  3. Prove that sin(x) / tan(x) = cos(x).

  4. Prove that cos(x) * tan(x) * csc(x) = 1.

  5. Prove that (1 - cos^2(x)) / sin(x) = sin(x).

  6. Prove that sin(2x) / (2cos(x)) = sin(x).

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Proving Trigonometric Identities Practice Sheet

Answer key

  1. Prove that tan(x) * cos(x) = sin(x).

    Start with LHS: tan(x)cos(x) = (sin(x)/cos(x)) * cos(x). Cancel cos(x) to get sin(x) = RHS.

  2. Prove that 1 - sin^2(x) = cos^2(x).

    Start with LHS: 1 - sin^2(x). Substitute 1 with sin^2(x) + cos^2(x). This gives sin^2(x) + cos^2(x) - sin^2(x) = cos^2(x) = RHS.

  3. Prove that sin(x) / tan(x) = cos(x).

    Start with LHS: sin(x) / (sin(x)/cos(x)) = sin(x) * (cos(x)/sin(x)). Cancel sin(x) to get cos(x) = RHS.

  4. Prove that cos(x) * tan(x) * csc(x) = 1.

    Start with LHS: cos(x) * (sin(x)/cos(x)) * (1/sin(x)). Multiply fractions to get (cos(x)sin(x)) / (cos(x)sin(x)) = 1 = RHS.

  5. Prove that (1 - cos^2(x)) / sin(x) = sin(x).

    Start with LHS: Replace 1 - cos^2(x) with sin^2(x). Then sin^2(x) / sin(x) = sin(x) = RHS.

  6. Prove that sin(2x) / (2cos(x)) = sin(x).

    Start with LHS: Apply the double-angle formula to the numerator: 2sin(x)cos(x) / (2cos(x)). Cancel 2 and cos(x) to get sin(x) = RHS.