Double Angle Simplification Quiz: Simplifying with Double Angle Formulas

  • 11th Grade
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Quizzes Created: 7682 | Total Attempts: 9,547,133
| Questions: 20 | Updated: Dec 16, 2025
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1) Simplify: sin2θ / (1 + cos2θ)

Explanation

Use sin2θ = 2sinθ cosθ and 1 + cos2θ = 2cos^2θ. Then sin2θ/(1 + cos2θ) = (2sinθ cosθ)/(2cos^2θ) = sinθ/cosθ = tanθ.

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About This Quiz
Double Angle Simplification Quiz: Simplifying With Double Angle Formulas - Quiz

Want to get better at simplifying trig expressions using double-angle formulas? In this quiz, you’ll work through problems that require selecting the right formula, rewriting expressions cleanly, and identifying patterns that lead to efficient simplification. You’ll analyze how double-angle identities transform complex terms into manageable forms and learn when each... see morevariation is most useful. By the end, you’ll feel more confident simplifying trigonometric expressions with clarity and strategic decision-making.
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2) Select all expressions equivalent to cos2θ (where defined)

Explanation

Standard forms: cos2θ = cos^2θ − sin^2θ = 2cos^2θ − 1 = 1 − 2sin^2θ. Also cos2θ = (1 − tan^2θ)/(1 + tan^2θ) when tanθ is defined. 2sin^2θ − 1 = −(1 − 2sin^2θ) ≠ cos2θ.

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3) Express cos^2θ in terms of cos2θ

Explanation

From cos2θ = 2cos^2θ − 1, we get 2cos^2θ = 1 + cos2θ ⇒ cos^2θ = (1 + cos2θ)/2.

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4) (1 + cos2θ)/2 equals sin^2θ.

Explanation

(1 + cos2θ)/2 equals cos^2θ, not sin^2θ. Therefore the statement is false.

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5) The identity (1 − cos2θ)/2 equals sin^2θ.

Explanation

From cos2θ = 1 − 2sin^2θ, rearrange: sin^2θ = (1 − cos2θ)/2. Hence the statement is true.

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6) Select all expressions equivalent to sinθ·cosθ

Explanation

Use sin2θ = 2sinθ cosθ ⇒ sinθ cosθ = sin2θ/2 = (1/2)·sin2θ. The others are not equal to sinθ cosθ.

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7) Simplify: 1 − cos2θ + 2cos^2θ

Explanation

Replace cos2θ with 2cos^2θ − 1: 1 − (2cos^2θ − 1) + 2cos^2θ = 1 − 2cos^2θ + 1 + 2cos^2θ = 2.

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8) Simplify: 2·sinθ·cosθ

Explanation

Use the double-angle identity sin2θ = 2sinθ cosθ. Therefore 2·sinθ·cosθ = sin2θ.

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9) Select all expressions equivalent to cos^2θ

Explanation

(1 + cos2θ)/2 = cos^2θ. Also 1 − sin^2θ = cos^2θ by Pythagorean identity. And 1 − (1 − cos2θ)/2 = (2 − 1 + cos2θ)/2 = (1 + cos2θ)/2 = cos^2θ. (1 − cos2θ)/2 = sin^2θ, and cos2θ ≠ cos^2θ.

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10) Simplify: cos^2θ − sin^2θ

Explanation

By the double-angle identity, cos2θ = cos^2θ − sin^2θ. The other listed forms are equivalent to cos2θ but the expression given simplifies directly to cos2θ.

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11) Simplify: sin2θ·cos2θ

Explanation

Use product-to-double-angle: sinα cosα = (1/2)sin(2α). With α = 2θ, sin2θ cos2θ = (1/2)sin4θ.

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12) Express sin^2θ − cos^2θ using cos2θ

Explanation

Since cos2θ = cos^2θ − sin^2θ, then sin^2θ − cos^2θ = −(cos^2θ − sin^2θ) = −cos2θ.

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13) 1 − cos2θ equals 2sin^2θ.

Explanation

From cos2θ = 1 − 2sin^2θ, rearrange to 1 − cos2θ = 2sin^2θ.

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14) Simplify: 4·sinθ·cosθ

Explanation

Since 2sinθ cosθ = sin2θ, multiply both sides by 2: 4sinθ cosθ = 2·(2sinθ cosθ) = 2sin2θ.

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15) Write sin^2θ·cos^2θ in terms of cos4θ

Explanation

First, sin^2θ cos^2θ = (1/4)sin^2(2θ). Then use sin^2α = (1 − cos2α)/2 with α = 2θ to get (1/4)·[(1 − cos4θ)/2] = (1 − cos4θ)/8.

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16) Tan(2θ) = (2tanθ)/(1 − tan^2θ)

Explanation

This is the double-angle identity for tangent, valid when 1 − tan^2θ ≠ 0 to avoid division by zero.

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17) Express sin^2θ in terms of cos2θ

Explanation

Start with cos2θ = 1 − 2sin^2θ. Solve for sin^2θ: 2sin^2θ = 1 − cos2θ ⇒ sin^2θ = (1 − cos2θ)/2.

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18) Simplify: 1 − 2sin^2θ

Explanation

One standard form is cos2θ = 1 − 2sin^2θ. So the expression simplifies to cos2θ.

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19) Select all expressions that equal sin^2θ + cos^2θ

Explanation

Identity: sin^2θ + cos^2θ = 1 (A). B simplifies to 1 since the cos2θ terms cancel. C uses the same identity at angle 2θ, equal to 1. E simplifies: 2·A − A + D? Compute: 2·[(1+cos2θ)/2] − [(1+cos2θ)/2] + [(1−cos2θ)/2] = [(1+cos2θ) − (1+cos2θ)/2 + (1−cos2θ)/2] = 1 after combining halves.

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20) Simplify: (1 − cos2θ) / (1 + cos2θ)

Explanation

Rewrite with power-reduction: 1 − cos2θ = 2sin^2θ and 1 + cos2θ = 2cos^2θ. The ratio is (2sin^2θ)/(2cos^2θ) = tan^2θ.

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Simplify: sin2θ / (1 + cos2θ)
Select all expressions equivalent to cos2θ (where defined)
Express cos^2θ in terms of cos2θ
(1 + cos2θ)/2 equals sin^2θ.
The identity (1 − cos2θ)/2 equals sin^2θ.
Select all expressions equivalent to sinθ·cosθ
Simplify: 1 − cos2θ + 2cos^2θ
Simplify: 2·sinθ·cosθ
Select all expressions equivalent to cos^2θ
Simplify: cos^2θ − sin^2θ
Simplify: sin2θ·cos2θ
Express sin^2θ − cos^2θ using cos2θ
1 − cos2θ equals 2sin^2θ.
Simplify: 4·sinθ·cosθ
Write sin^2θ·cos^2θ in terms of cos4θ
Tan(2θ) = (2tanθ)/(1 − tan^2θ)
Express sin^2θ in terms of cos2θ
Simplify: 1 − 2sin^2θ
Select all expressions that equal sin^2θ + cos^2θ
Simplify: (1 − cos2θ) / (1 + cos2θ)
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