Even–Odd Trig Identities: Apply & Simplify Quiz

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| By Thames
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Quizzes Created: 7116 | Total Attempts: 9,522,086
| Questions: 20 | Updated: Oct 31, 2025
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1) Simplify using even/odd: sin(−x) + sin(x).

Explanation

sin(−x)=−sin x ⇒ −sin x + sin x = 0.

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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

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

Explanation

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.

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3) Which identity is true for all real x?

Explanation

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).

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4) Simplify: 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.

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5) Evaluate: sin(−π/6) + cos(−π/3).

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.

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

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7) Simplify: 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.

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8) Simplify: 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.

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

Explanation

Given: supplementary angle. Goal: simplify.

Step 1: sin(π − x) = sin x.

So, the final answer is sin x.

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10) Which is an even function?

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.

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

Explanation

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).

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12) Simplify: sin(−x) cos(−x) − 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.

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13) Simplify completely: cos(−x) + 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.

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

Explanation

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

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15) Which statement is true?

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.

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

Explanation

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.

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

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.

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18) Evaluate: tan(−π/4).

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.

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19) Simplify using identities: 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.

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

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.

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Simplify using even/odd: sin(−x) + sin(x).
Simplify completely: cos(−θ) − sin(−θ).
Which identity is true for all real x?
Simplify: cos(−x) − sin(x).
Evaluate: sin(−π/6) + cos(−π/3).
Simplify using properties: tan(−x) · tan(x).
Simplify: tan(−x) + cot(−x).
Simplify: sin(−x) cos(−x).
Use even/odd identities to simplify: sin(π − x).
Which is an even function?
Evaluate using parity: sec(−π/4) + csc(−π/4).
Simplify: sin(−x) cos(−x) − sin x cos x.
Simplify completely: cos(−x) + cos x.
Simplify using properties: tan(−x) − tan x.
Which statement is true?
Simplify: csc(−x) + sec(−x) − [csc x − sec x].
Simplify: sin(−x) + sin x + cos(−x) − cos x.
Evaluate: tan(−π/4).
Simplify using identities: sin(−x) cos(−x) = ?.
Solve using symmetry and reference angles: tan(−5π/6).
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