Even–Odd Trig Identities: Apply & Simplify Quiz

Grade 11th

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| Questions: 20 | Updated: Jan 22, 2026

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1) 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

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.
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2)

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2) 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

3) 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

4) 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

5) 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

6) 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

7) 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

8) 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

9) 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

10) 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

11) 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

12) 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

13) 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

14) 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

15) 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

16) 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

17) 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

18) 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

19) 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

20) 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