Even–Odd Trig Identities: Apply & Simplify Quiz

11th Grade

Reviewed by Cierra Henderson

Cierra Henderson, MBA |

K-12 Expert

Review Board Member

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.

, MBA

By Thames

Thames

Community Contributor

Quizzes Created: 8156 | Total Attempts: 9,586,195

| Questions: 20 | Updated: Jan 22, 2026

Share on Facebook Share on Twitter Share on Whatsapp Share on Pinterest Share on Email Copy to Clipboard Embed on your website

Question 1 / 21

🏆 Rank #-- ▾

0 %

0/100

Score 0/100

1) Simplify using even/odd: sin(−x) + sin(x).

2 sin x

2 cos x

sin(−x)=−sin x ⇒ −sin x + sin x = 0.

Explanation

sin(−x)=−sin x ⇒ −sin x + sin x = 0.

Submit

About This Quiz

Evenodd Trig Identities: Apply & Simplify Quiz - Quiz

Can you simplify tricky trig expressions using symmetry rules? In this quiz, you’ll apply the even–odd identities to expressions that mix positive and negative angles. You’ll combine, compare, and simplify terms like sin(−x) + sin(x) or cos(−x) − cos(x), and practice evaluating exact values using symmetry. Step by step, you’ll... see morelearn how even–odd relationships make trigonometric simplification faster and more intuitive.
see less

2) What first name or nickname would you like us to use?

You may optionally provide this to label your report, leaderboard, or certificate.

2) Simplify completely: cos(−θ) − sin(−θ).

−cos θ − sin θ

Cos θ + sin θ

Cos θ − (−sin θ)

Cos θ + (−sin θ)

Given: cos(−θ) − sin(−θ). Goal: simplify.

Step 1: cos even ⇒ cos(−θ)=cos θ.

Step 2: sin odd ⇒ sin(−θ)=−sin θ.

Step 3: cos θ − (−sin θ) = cos θ + sin θ.

So, the final answer is cos θ + sin θ.

Explanation

Submit

3) Which identity is true for all real x?

Tan(−x)=tan(x)

Sec(−x)=−sec(x)

Csc(−x)=−csc(x)

Cot(−x)=−cot(x)

Given: parity. Goal: choose one true identity.

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

Step 2: A is false; B is false; C is true (also D true, but one choice suffices).

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

Explanation

Submit

4) Simplify: cos(−x) − sin(x).

Cos x − sin x

−cos x − sin x

Cos x − sin x

Cos x + sin x

Given: cos(−x)−sin x. Goal: simplify.

Step 1: cos even ⇒ cos(−x)=cos x.

Step 2: Expression becomes cos x − sin x.

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

Explanation

Given: cos(−x)−sin x. Goal: simplify.

Step 1: cos even ⇒ cos(−x)=cos x.

Step 2: Expression becomes cos x − sin x.

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

Submit

5) Evaluate: sin(−π/6) + cos(−π/3).

−1/2 + 1/2

1/2 + 1/2

−1/2 + 1

−1/2 + 1/2√3

Given: exact values. Goal: evaluate form.

Step 1: sin(−π/6)=−1/2.

Step 2: cos(−π/3)=cos(π/3)=1/2.

Step 3: Sum = −1/2 + 1/2 = 0 (numerically).

So, the final answer is −1/2 + 1/2.

Explanation

Given: exact values. Goal: evaluate form.

Step 1: sin(−π/6)=−1/2.

Step 2: cos(−π/3)=cos(π/3)=1/2.

Step 3: Sum = −1/2 + 1/2 = 0 (numerically).

So, the final answer is −1/2 + 1/2.

Submit

6) Simplify using properties: tan(−x) · tan(x).

−tan² x

Tan² x

Given: product of tangents. Goal: simplify.

Step 1: tan(−x)=−tan x.

Step 2: (−tan x)(tan x)=−tan² x.

So, the final answer is −tan² x.

Explanation

Given: product of tangents. Goal: simplify.

Step 1: tan(−x)=−tan x.

Step 2: (−tan x)(tan x)=−tan² x.

So, the final answer is −tan² x.

Submit

7) Simplify: tan(−x) + cot(−x).

Tan x + cot x

−tan x − cot x

−tan x + cot x

Tan x − cot x

Given: sum. Goal: simplify.

Step 1: tan odd ⇒ −tan x; cot odd ⇒ −cot x.

Step 2: Sum = −tan x − cot x.

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

Explanation

Given: sum. Goal: simplify.

Step 1: tan odd ⇒ −tan x; cot odd ⇒ −cot x.

Step 2: Sum = −tan x − cot x.

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

Submit

8) Simplify: sin(−x) cos(−x).

Sin x cos x

−sin x cos x

Sin² x − cos² x

Given: product. Goal: simplify.

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

Step 2: Product = −sin x cos x.

So, the final answer is −sin x cos x.

Explanation

Given: product. Goal: simplify.

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

Step 2: Product = −sin x cos x.

So, the final answer is −sin x cos x.

Submit

9) Use even/odd identities to simplify: sin(π − x).

Sin x

−sin x

Cos x

−cos x

Given: supplementary angle. Goal: simplify.

Step 1: sin(π − x) = sin x.

So, the final answer is sin x.

Explanation

Given: supplementary angle. Goal: simplify.

Step 1: sin(π − x) = sin x.

So, the final answer is sin x.

Submit

10) Which is an even function?

Sin x + cos x

Sin x cos x

Cos² x

Tan x

Given: parity check. Goal: choose an even function.

Step 1: cos is even; squaring preserves evenness ⇒ cos² x is even.

So, the final answer is cos² x.

Explanation

Given: parity check. Goal: choose an even function.

Step 1: cos is even; squaring preserves evenness ⇒ cos² x is even.

So, the final answer is cos² x.

Submit

11) Evaluate using parity: sec(−π/4) + csc(−π/4).

√2 + √2

√2 − √2

√2 + (−√2)

−√2 − √2

Given: sec even; csc odd. Goal: compute.

Step 1: sec(−π/4)=sec(π/4)=√2.

Step 2: csc(−π/4)=−csc(π/4)=−√2.

Step 3: Sum = √2 + (−√2) = 0.

So, the final answer is √2 + (−√2).

Explanation

Submit

12) Simplify: sin(−x) cos(−x) − sin x cos x.

−2 sin x cos x

2 sin x cos x

−sin 2x

Given: difference. Goal: simplify.

Step 1: sin(−x)cos(−x)=−sin x cos x.

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

So, the final answer is −2 sin x cos x.

Explanation

Given: difference. Goal: simplify.

Step 1: sin(−x)cos(−x)=−sin x cos x.

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

So, the final answer is −2 sin x cos x.

Submit

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

2 cos x

−2 cos x

Cos² x

Given: sum with cos(−x). Goal: simplify.

Step 1: cos even ⇒ cos(−x)=cos x.

Step 2: Sum = cos x + cos x = 2 cos x.

So, the final answer is 2 cos x.

Explanation

Given: sum with cos(−x). Goal: simplify.

Step 1: cos even ⇒ cos(−x)=cos x.

Step 2: Sum = cos x + cos x = 2 cos x.

So, the final answer is 2 cos x.

Submit

14) Simplify using properties: tan(−x) − tan x.

−2 tan x

2 tan x

Tan² x

tan(−x)=−tan x ⇒ (−tan x)−tan x=−2 tan x.

Explanation

tan(−x)=−tan x ⇒ (−tan x)−tan x=−2 tan x.

Submit

15) Which statement is true?

Cos(−x)=−cos x; sin(−x)=sin x

Cos(−x)=cos x; sin(−x)=−sin x

Cos(−x)=−cos x; sin(−x)=−sin x

Cos(−x)=cos x; sin(−x)=sin x

Goal: Correct statement

Step 1: cosine is even; sine is odd.

So, the final answer is cos(−x)=cos x; sin(−x)=−sin x.

Explanation

Goal: Correct statement

Step 1: cosine is even; sine is odd.

So, the final answer is cos(−x)=cos x; sin(−x)=−sin x.

Submit

16) Simplify: csc(−x) + sec(−x) − [csc x − sec x].

−2 csc x

−2 sec x

−2 csc x + 2 sec x

Given: expression. Goal: simplify.

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

Step 2: Left part = −csc x + sec x.

Step 3: Subtract (csc x − sec x): (−csc x + sec x) − (csc x − sec x) = −2 csc x + 2 sec x.

So, the final answer is −2 csc x + 2 sec x.

Explanation

Submit

17) Simplify: sin(−x) + sin x + cos(−x) − cos x.

2 sin x

2 cos x

Given: combined sum. Goal: simplify.

Step 1: sin(−x) = −sin x ⇒ cancels with +sin x.

Step 2: cos(−x) = cos x ⇒ cos x − cos x cancels.

So, the final answer is 0.

Explanation

Given: combined sum. Goal: simplify.

Step 1: sin(−x) = −sin x ⇒ cancels with +sin x.

Step 2: cos(−x) = cos x ⇒ cos x − cos x cancels.

So, the final answer is 0.

Submit

18) Evaluate: tan(−π/4).

−1

√3

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

Step 1: Tangent is odd ⇒ tan(−π/4) = −tan(π/4).

Step 2: tan(π/4) = 1 ⇒ result = −1.

So, the final answer is −1.

Explanation

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

Step 1: Tangent is odd ⇒ tan(−π/4) = −tan(π/4).

Step 2: tan(π/4) = 1 ⇒ result = −1.

So, the final answer is −1.

Submit

19) Simplify using identities: sin(−x) cos(−x) = ?.

−sin x cos x

Sin x cos x

Sin x

Cos x

Given: product under parity. Goal: simplify.

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

Step 2: Product = −sin x cos x.

So, the final answer is −sin x cos x.

Explanation

Given: product under parity. Goal: simplify.

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

Step 2: Product = −sin x cos x.

So, the final answer is −sin x cos x.

Submit

20) Solve using symmetry and reference angles: tan(−5π/6).

−√3

√3/3

−√3/3

√3

Given: −5π/6. Goal: value of tangent.

Step 1: tan(−θ) = −tan θ.

Step 2: tan(5π/6) = −√3/3 (QII).

Step 3: −(−√3/3) = √3/3.

So, the final answer is √3/3.

Explanation

Given: −5π/6. Goal: value of tangent.

Step 1: tan(−θ) = −tan θ.

Step 2: tan(5π/6) = −√3/3 (QII).

Step 3: −(−√3/3) = √3/3.

So, the final answer is √3/3.

Submit