Even Odd Identities Quiz: Fundamental Even Odd Definitions

  • 11th Grade
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| Questions: 20 | Updated: Dec 16, 2025
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1) Which expression is even in θ?

Explanation

sin is odd, but squaring removes the sign: sin^2(−θ)=(sin(−θ))^2=(−sinθ)^2=sin^2θ, so it is even.

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About This Quiz
Even Odd Identities Quiz: Fundamental Even Odd Definitions - Quiz

What determines whether a trigonometric function is even or odd? In this quiz, you’ll explore how symmetry in the coordinate plane shapes the behavior of trig graphs and their algebraic rules. You’ll analyze reflections across axes, observe sign changes, and connect these visual patterns to symbolic definitions. With each question,... see moreyou’ll build stronger intuition for how functions respond to negative inputs and why these identities become essential tools in simplifying trig expressions and verifying equations.
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2) Classify cosθ by parity.

Explanation

cos(−θ)=cosθ for all θ, so cosine is even by definition of even functions.

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3) Simplify csc(−θ).

Explanation

cscθ=1/sinθ and sine is odd, so csc(−θ)=1/sin(−θ)=1/(−sinθ)=−cscθ.

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4) Simplify cot(−θ).

Explanation

cotθ=cosθ/sinθ with cos even and sin odd, so cot(−θ)=cos(−θ)/sin(−θ)=cosθ/(−sinθ)=−cotθ.

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5) Simplify sin(−θ)·cos(−θ).

Explanation

sin is odd and cos is even. So sin(−θ)·cos(−θ)=(−sinθ)(cosθ)=−(sinθ·cosθ).

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6) Simplify cos(−θ).

Explanation

Cosine is even: cos(−θ)=cosθ because x-coordinates on the unit circle are unchanged under reflection across the x-axis.

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7) Sec(−θ)=−secθ.

Explanation

secθ=1/cosθ and cosine is even. So sec(−θ)=1/cos(−θ)=1/cosθ=secθ, not −secθ.

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8) Select all identities that are always true.

Explanation

cos and sec are even so they are unchanged under −θ. tan, sin, and cot are odd, so they change sign under −θ.

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9) Simplify sin(−θ).

Explanation

Sine is odd: sin(−θ)=−sinθ because reflecting θ across the x-axis flips the y-coordinate on the unit circle.

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10) Evaluate tan(−θ)/tanθ (when defined).

Explanation

Since tan(−θ)=−tanθ, the ratio tan(−θ)/tanθ=(−tanθ)/tanθ=−1 whenever tanθ≠0.

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11) The quotient of two odd functions (where defined) is even.

Explanation

If f and g are odd, (f/g)(−x)=f(−x)/g(−x)=(−f(x))/(−g(x))=f(x)/g(x). Thus the quotient is even when the denominator is nonzero.

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12) If f is even and g is odd, then f·g is odd.

Explanation

Let f(−x)=f(x) and g(−x)=−g(x). Then (fg)(−x)=f(−x)g(−x)=f(x)(−g(x))=−f(x)g(x), so fg is odd.

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13) Select all statements that are always true.

Explanation

A: cos is even. B: odd power of an odd function stays odd, so sin^3(−θ)=−sin^3θ. C: tan^2 is even, so the given negative is false. D: sin is odd and cos is even, so the product is odd, not even. E: csc is odd.

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14) Select all true statements using exact values at 30°.

Explanation

At 30°: sin30°=1/2, cos30°=√3/2, tan30°=1/√3. Using parity: sin is odd (becomes −1/2), cos is even (stays √3/2), tan is odd (becomes −1/√3), sec is even (unchanged). csc is odd so it changes sign.

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15) The product of two odd functions is even.

Explanation

If f and g are odd, f(−x)=−f(x), g(−x)=−g(x). Then (fg)(−x)=f(−x)g(−x)=(−f)(−g)=fg, so fg is even.

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16) Simplify sec(−θ).

Explanation

secθ=1/cosθ and cos is even, so sec(−θ)=1/cos(−θ)=1/cosθ=secθ.

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17) Select all functions that are odd.

Explanation

sin, tan, cot, and csc are odd: f(−θ)=−f(θ). cos and sec are even: f(−θ)=f(θ).

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18) Tan(−θ)=−tanθ.

Explanation

tanθ=sinθ/cosθ with sin odd and cos even. Then tan(−θ)=sin(−θ)/cos(−θ)=(−sinθ)/(cosθ)=−tanθ.

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19) Simplify −cos(−θ).

Explanation

cos is even, so cos(−θ)=cosθ. Therefore −cos(−θ)=−cosθ.

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20) On the unit circle, (x,y)=(cosθ,sinθ). Write tan(−θ) in terms of x and y.

Explanation

tanθ=y/x. Since tan is odd, tan(−θ)=−tanθ=−(y/x)=−y/x (x≠0).

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Which expression is even in θ?
Classify cosθ by parity.
Simplify csc(−θ).
Simplify cot(−θ).
Simplify sin(−θ)·cos(−θ).
Simplify cos(−θ).
Sec(−θ)=−secθ.
Select all identities that are always true.
Simplify sin(−θ).
Evaluate tan(−θ)/tanθ (when defined).
The quotient of two odd functions (where defined) is even.
If f is even and g is odd, then f·g is odd.
Select all statements that are always true.
Select all true statements using exact values at 30°.
The product of two odd functions is even.
Simplify sec(−θ).
Select all functions that are odd.
Tan(−θ)=−tanθ.
Simplify −cos(−θ).
On the unit circle, (x,y)=(cosθ,sinθ). Write tan(−θ) in terms of...
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