Trigonometric Simplification Quiz: Simplifying Trigonometric Expressions

  • 10th Grade
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| Questions: 20 | Updated: Dec 16, 2025
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1) Simplify: sin^2θ + cos^2θ

Explanation

Use the Pythagorean identity sin^2θ + cos^2θ = 1. No matter the value of θ, the sum of the squares of sine and cosine equals 1.

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About This Quiz
Trigonometric Simplification Quiz: Simplifying Trigonometric Expressions - Quiz

Ready to clean up tricky trig expressions? This quiz helps you simplify using identities, substitutions, and basic relationships. You’ll break expressions down, spot shortcuts, and see how small steps can make a big difference. Explore the questions and strengthen your trig-simplifying skills.

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2) Simplify: sec^2θ − tan^2θ

Explanation

From 1 + tan^2θ = sec^2θ, rearrange to sec^2θ − tan^2θ = 1.

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3) Simplify: csc^2θ − 1

Explanation

From 1 + cot^2θ = csc^2θ, subtract 1 to get csc^2θ − 1 = cot^2θ.

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4) Simplify: (1 + tan^2θ) / sec^2θ

Explanation

Since 1 + tan^2θ = sec^2θ, the expression becomes sec^2θ/sec^2θ = 1.

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5) Simplify: (1 + cot^2θ) − csc^2θ

Explanation

Since 1 + cot^2θ = csc^2θ, the expression equals csc^2θ − csc^2θ = 0.

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6) Select all expressions equivalent to sin^2θ

Explanation

sin^2θ = 1 − cos^2θ by sin^2θ + cos^2θ = 1. Also, 1/csc^2θ = sin^2θ since csc^2θ = 1/sin^2θ. The others simplify to 1 or are not identities for sin^2θ.

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7) Select all expressions equivalent to cot^2θ

Explanation

cot^2θ = csc^2θ − 1 and cot^2θ = cos^2θ/sin^2θ = 1/tan^2θ. (1 − cos^2θ)/sin^2θ = sin^2θ/sin^2θ = 1. sec^2θ − 1 = tan^2θ.

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8) Simplify: (1 − sin^2θ) / cos^2θ

Explanation

1 − sin^2θ = cos^2θ. Therefore (1 − sin^2θ)/cos^2θ = cos^2θ/cos^2θ = 1.

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9) 1 − sin^2θ equals cos^2θ

Explanation

From sin^2θ + cos^2θ = 1, subtract sin^2θ to get 1 − sin^2θ = cos^2θ.

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10) Sec^2θ equals 1 − tan^2θ

Explanation

The identity is 1 + tan^2θ = sec^2θ. Therefore sec^2θ = 1 + tan^2θ, not 1 − tan^2θ.

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11) Select all expressions equal to 1

Explanation

By the Pythagorean identities: sin^2θ + cos^2θ = 1, sec^2θ − tan^2θ = 1, csc^2θ − cot^2θ = 1. Also, (1 − sin^2θ)/cos^2θ = cos^2θ/cos^2θ = 1. tan^2θ − sec^2θ = −1.

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12) Csc^2θ − cot^2θ equals 1

Explanation

From 1 + cot^2θ = csc^2θ, rearrange to csc^2θ − cot^2θ = 1.

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13) Simplify: (sec^2θ − 1) / tan^2θ

Explanation

sec^2θ − 1 = tan^2θ. So (sec^2θ − 1)/tan^2θ = tan^2θ/tan^2θ = 1.

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14) Tan^2θ + 1 equals csc^2θ

Explanation

tan^2θ + 1 = sec^2θ. csc^2θ relates to cot^2θ by 1 + cot^2θ = csc^2θ.

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15) Simplify: sin^2θ / (1 − cos^2θ)

Explanation

1 − cos^2θ = sin^2θ. Thus sin^2θ/(1 − cos^2θ) = sin^2θ/sin^2θ = 1.

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16) Simplify: (1 − sin^2θ) / (1 − cos^2θ)

Explanation

1 − sin^2θ = cos^2θ and 1 − cos^2θ = sin^2θ. Therefore (1 − sin^2θ)/(1 − cos^2θ) = cos^2θ/sin^2θ = cot^2θ.

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17) Select all expressions equivalent to sec^2θ

Explanation

sec^2θ = 1 + tan^2θ and sec^2θ = 1/cos^2θ. csc^2θ − cot^2θ = 1. 1 − tan^2θ is not an identity. (1 + sin^2θ)/cos^2θ = sec^2θ + tan^2θ, not just sec^2θ.

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18) Simplify: (sec^2θ − 1) / (1 + tan^2θ)

Explanation

sec^2θ − 1 = tan^2θ and 1 + tan^2θ = sec^2θ. So the fraction is tan^2θ/sec^2θ = (sin^2θ/cos^2θ)/(1/cos^2θ) = sin^2θ.

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19) Write tan^2θ in terms of sec^2θ

Explanation

From 1 + tan^2θ = sec^2θ, rearrange to tan^2θ = sec^2θ − 1.

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20) Express 1 − cos^2θ in terms of sine only

Explanation

Since sin^2θ + cos^2θ = 1, then 1 − cos^2θ = sin^2θ.

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Simplify: sin^2θ + cos^2θ
Simplify: sec^2θ − tan^2θ
Simplify: csc^2θ − 1
Simplify: (1 + tan^2θ) / sec^2θ
Simplify: (1 + cot^2θ) − csc^2θ
Select all expressions equivalent to sin^2θ
Select all expressions equivalent to cot^2θ
Simplify: (1 − sin^2θ) / cos^2θ
1 − sin^2θ equals cos^2θ
Sec^2θ equals 1 − tan^2θ
Select all expressions equal to 1
Csc^2θ − cot^2θ equals 1
Simplify: (sec^2θ − 1) / tan^2θ
Tan^2θ + 1 equals csc^2θ
Simplify: sin^2θ / (1 − cos^2θ)
Simplify: (1 − sin^2θ) / (1 − cos^2θ)
Select all expressions equivalent to sec^2θ
Simplify: (sec^2θ − 1) / (1 + tan^2θ)
Write tan^2θ in terms of sec^2θ
Express 1 − cos^2θ in terms of sine only
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