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Math 30-1 · Worksheets

Trig identities · Set B

Ten questions of mixed difficulty, covering Trig identities. Print it, or work through it on screen — the answer key starts on its own page.

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Trig identities · Set B

Math 30-1 · maddyhelps.com

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Circle the best answer for each question. Show your work in the space provided.

  1. Simplify (1 − sin²x) ÷ cos x.

    1. a) sec x
    2. b) 1
    3. c) cos x
    4. d) sin x

  2. Which expression is equal to sin 2x?

    1. a) 2 sin x
    2. b) sin x + cos x
    3. c) sin²x − cos²x
    4. d) 2 sin x cos x

  3. sin²x + cos²x equals:

    1. a) tan²x
    2. b) 1
    3. c) 0
    4. d) sec²x

  4. What is the exact value of 2 sin 15° cos 15°?

    1. a) √2/2
    2. b) 1/4
    3. c) √3/2
    4. d) 1/2

  5. When proving a trigonometric identity, you should:

    1. a) substitute one value of x and check both sides
    2. b) work on each side separately until both sides match
    3. c) move terms from one side to the other, as when solving an equation
    4. d) square both sides first

  6. Using a sum identity, what is the exact value of cos 105°?

    1. a) −1/2
    2. b) (√2 − √6)/4
    3. c) (√6 − √2)/4
    4. d) (√2 + √6)/4

  7. Which expression is equal to tan x?

    1. a) 1/sin x
    2. b) sin x/cos x
    3. c) sin x · cos x
    4. d) cos x/sin x

  8. What is the exact value of cos²(π/8) − sin²(π/8)?

    1. a) √2/2
    2. b) √3/2
    3. c) 1/2
    4. d) 1

  9. Substituting x = π/4 makes both sides of an equation equal. What does this show?

    1. a) That the equation is an identity
    2. b) Only that the equation is true for x = π/4 — it does not prove an identity
    3. c) That the equation has no non-permissible values
    4. d) That the equation is never true

  10. What is the non-permissible value of sin x ÷ (1 − cos x) for 0 ≤ x < 2π?

    1. a) x = π
    2. b) x = 0
    3. c) x = π/2 and x = 3π/2
    4. d) there is none

Answer key · Trig identities · Set B

Math 30-1 · maddyhelps.com

  1. c) cos x — 1 − sin²x = cos²x from the Pythagorean identity, and cos²x ÷ cos x = cos x.
  2. d) 2 sin x cos x — The double-angle identity is sin 2x = 2 sin x cos x. Doubling the angle does not double the sine — 2 sin x is a different thing.
  3. b) 1 — On the unit circle, sin x and cos x are the legs of a right triangle with hypotenuse 1, so the Pythagorean theorem gives sin²x + cos²x = 1.
  4. d) 1/2 — Recognise the double-angle pattern: 2 sin 15° cos 15° = sin 30° = 1/2. Spotting the identity saves any calculation.
  5. b) work on each side separately until both sides match — An identity is what you are trying to prove, so you cannot treat it as a true equation yet. Simplify each side on its own until they are identical.
  6. b) (√2 − √6)/4 — cos(60° + 45°) = cos 60° cos 45° − sin 60° sin 45° = (1/2)(√2/2) − (√3/2)(√2/2) = (√2 − √6)/4. It is negative, as it must be in Quadrant II.
  7. b) sin x/cos x — tan x = y/x on the unit circle, and y = sin x while x = cos x. cos x/sin x is cot x.
  8. a) √2/2 — cos²A − sin²A = cos 2A, so this is cos(π/4) = √2/2. Do not confuse it with cos²A + sin²A, which is 1.
  9. b) Only that the equation is true for x = π/4 — it does not prove an identity — An identity must hold for every permissible value. One matching value, or even several, can happen by coincidence. Only an algebraic proof shows it is always true.
  10. b) x = 0 — The denominator is zero when cos x = 1, which happens at x = 0 in this domain. In general, x ≠ 2πn.