Even–Odd Trig Identities: Proof & Verification Quiz

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Cierra Henderson, MBA |
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Cierra is an educational consultant and curriculum developer who has worked with students in K-12 for a variety of subjects including English and Math as well as test prep. She specializes in one-on-one support for students especially those with learning differences. She holds an MBA from the University of Massachusetts Amherst and a certificate in educational consulting from UC Irvine.
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| Attempts: 11 | Questions: 20 | Updated: Dec 11, 2025
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1) Which statement is always true?

Explanation

Given: even/odd properties. Goal: pick the always-true statement.

Step 1: Sine is odd, cosine is even, tangent is odd, secant is even.

Step 2: For an odd function f, f(−x) = −f(x); for an even function g, g(−x) = g(x).

Step 3: Therefore tan(−x) = −tan(x) is always true.

So, the final answer is tan(−x) = −tan(x).

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About This Quiz
Evenodd Trig Identities: Proof & Verification Quiz - Quiz

Are you ready to discover how flipping angles affects trig functions? In this quiz, you’ll explore how sine, cosine, and tangent respond when you replace θ with −θ. You’ll test and verify that sine and tangent are odd functions while cosine is even — all through examples, evaluations, and simple... see moreproofs. By the end, you’ll see how symmetry on the coordinate plane helps explain why these patterns always hold true!
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2) Which is an even function?

Explanation

Given: parity of basic trig functions. Goal: identify the even one.

Step 1: cos is even; sin, tan, csc are odd on their domains.

So, the final answer is cos x.

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3) If sin x = 5/13, what is sin(−x)?

Explanation

Given: sin x = 5/13. Goal: find sin(−x).

Step 1: Sine is odd ⇒ sin(−x) = −sin x.

Step 2: Substitute sin x = 5/13 ⇒ sin(−x) = −5/13.

So, the final answer is −5/13.

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4) Evaluate cos(−π/3).

Explanation

Given: cos(−θ). Goal: evaluate at θ = π/3.

Step 1: Cosine is even ⇒ cos(−π/3) = cos(π/3).

Step 2: cos(π/3) = 1/2.

So, the final answer is 1/2.

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5) Which identity is correct?

Explanation

Given: parity of reciprocal functions. Goal: choose a true identity.

Step 1: sec is even; csc and cot are odd; tan is odd.

Step 2: Therefore sec(−x) = sec(x) is correct.

So, the final answer is sec(−x) = sec(x).

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6) If tan θ = −2, what is tan(−θ)?

Explanation

Given: tan θ = −2. Goal: compute tan(−θ).

Step 1: Tangent is odd ⇒ tan(−θ) = −tan θ.

Step 2: −(−2) = 2.

So, the final answer is 2.

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7) Simplify sin(−7π/6).

Explanation

Given: angle −7π/6. Goal: value of sine.

Step 1: Sine is odd ⇒ sin(−7π/6) = −sin(7π/6).

Step 2: sin(7π/6) = −1/2 (QIII).

Step 3: −(−1/2) = 1/2.

So, the final answer is 1/2.

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8) Simplify and classify f(x) = sin x · cos x.

Explanation

Given: product sin x cos x. Goal: classify parity.

Step 1: sin is odd, cos is even.

Step 2: Odd × Even = Odd.

So, the final answer is Odd.

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9) Simplify using even/odd: sec(−x) + tan(−x).

Explanation

Given: sec(−x)+tan(−x). Goal: simplify.

Step 1: sec even ⇒ sec(−x)=sec x; tan odd ⇒ tan(−x)=−tan x.

Step 2: Sum = sec x − tan x.

So, the final answer is sec x − tan x.

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10) Evaluate sin(−π/4).

Explanation

Given: sin(−θ). Goal: evaluate at θ = π/4.

Step 1: Sine is odd ⇒ sin(−π/4) = −sin(π/4).

Step 2: sin(π/4) = √2/2 ⇒ result = −√2/2.

So, the final answer is −√2/2.

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11) Simplify: cos(−x) + sin(−x).

Explanation

Given: cos(−x)+sin(−x). Goal: simplify.

Step 1: cos even ⇒ cos(−x)=cos x.

Step 2: sin odd ⇒ sin(−x)=−sin x.

Step 3: Sum = cos x − sin x.

So, the final answer is cos x − sin x.

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12) If cos θ = −3/5, what is cos(−θ)?

Explanation

Given: cos θ = −3/5. Goal: cos(−θ).

Step 1: Cosine is even ⇒ cos(−θ) = cos θ.

Step 2: Therefore cos(−θ) = −3/5.

So, the final answer is −3/5.

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13) Simplify completely: sec(−x) − cos(−x).

Explanation

Given: sec(−x)−cos(−x). Goal: simplify.

Step 1: sec even ⇒ sec(−x)=sec x; cos even ⇒ cos(−x)=cos x.

Step 2: Difference = sec x − cos x.

So, the final answer is sec x − cos x.

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14) Evaluate tan(−π/3).

Explanation

Given: tan(−θ). Goal: evaluate at θ = π/3.

Step 1: Tangent is odd ⇒ tan(−π/3)=−tan(π/3).

Step 2: tan(π/3)=√3 ⇒ result = −√3.

So, the final answer is −√3.

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15) Simplify: tan(−x) · cot(−x).

Explanation

Given: product of tangent and cotangent at −x. Goal: simplify.

Step 1: tan(−x)=−tan x; cot(−x)=−cot x.

Step 2: Product = (−tan x)(−cot x)=tan x·cot x=1 (where defined).

So, the final answer is 1.

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16) The graph of y = sin x is symmetric with respect to the

Explanation

Given: symmetry of sine. Goal: identify the axis/point.

Step 1: Sine is odd ⇒ origin symmetry.

So, the final answer is origin (odd).

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17) If g(x) = sec x + tan x, then g(−x) equals:

Explanation

Given: g(x)=sec x + tan x. Goal: g(−x).

Step 1: sec even ⇒ sec(−x)=sec x; tan odd ⇒ tan(−x)=−tan x.

Step 2: g(−x) = sec x − tan x.

So, the final answer is sec x − tan x.

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18) Simplify using even/odd: cos(−x) · sin(−x).

Explanation

Given: product. Goal: simplify.

Step 1: cos even ⇒ cos(−x)=cos x; sin odd ⇒ sin(−x)=−sin x.

Step 2: Product = cos x · (−sin x) = −cos x · sin x.

So, the final answer is −cos x · sin x.

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19) If f is even and g is odd, which is always odd?

Explanation

Given: parity rules. Goal: pick an odd result.

Step 1: Even × Odd = Odd.

Step 2: Therefore f(x)g(x) is odd.

So, the final answer is f(x)g(x).

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20) Evaluate csc(−π/6).

Explanation

Given: csc(−θ). Goal: evaluate at θ = π/6.

Step 1: csc is odd ⇒ csc(−π/6) = −csc(π/6).

Step 2: csc(π/6) = 2 ⇒ result = −2.

So, the final answer is −2.

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Cierra Henderson |MBA |
K-12 Expert
Cierra is an educational consultant and curriculum developer who has worked with students in K-12 for a variety of subjects including English and Math as well as test prep. She specializes in one-on-one support for students especially those with learning differences. She holds an MBA from the University of Massachusetts Amherst and a certificate in educational consulting from UC Irvine.
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Which statement is always true?
Which is an even function?
If sin x = 5/13, what is sin(−x)?
Evaluate cos(−π/3).
Which identity is correct?
If tan θ = −2, what is tan(−θ)?
Simplify sin(−7π/6).
Simplify and classify f(x) = sin x · cos x.
Simplify using even/odd: sec(−x) + tan(−x).
Evaluate sin(−π/4).
Simplify: cos(−x) + sin(−x).
If cos θ = −3/5, what is cos(−θ)?
Simplify completely: sec(−x) − cos(−x).
Evaluate tan(−π/3).
Simplify: tan(−x) · cot(−x).
The graph of y = sin x is symmetric with respect to the
If g(x) = sec x + tan x, then g(−x) equals:
Simplify using even/odd: cos(−x) · sin(−x).
If f is even and g is odd, which is always odd?
Evaluate csc(−π/6).
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