Master Trig Identities: Ace Your College Algebra Exams

Question 1

Which trigonometric identity is frequently used to simplify expressions involving the product of sine and cosine of different angles?

Accepted Answer

sin(A + B) = sin A cos B + cos A sin B

Answer

cos(A + B) = cos A cos B - sin A sin B

Answer

tan(A + B) = (tan A + tan B) / (1 - tan A tan B)

Answer

cot(A + B) = (cot A cot B - 1) / (cot A + cot B)

Question 2

From the given options, select the correct identity for sin(2A) expressed in terms of sin A and cos A.

Accepted Answer

sin(2A) = 2sin A cos A

Answer

sin(2A) = sin^2 A - cos^2 A

Answer

sin(2A) = cos^2 A - sin^2 A

Answer

sin(2A) = 2cos A - 2sin A

Question 3

In trigonometry, which identity serves as the foundation for proving the Pythagorean identity?

Accepted Answer

sin^2 A + cos^2 A = 1

Answer

1 + tan^2 A = sec^2 A

Answer

cos^2 A - sin^2 A = tan^2 A

Answer

csc^2 A - cot^2 A = 1

Question 4

Identify the correct identity for cos(A/2) expressed in terms of sin A.

Accepted Answer

cos(A/2) = sqrt((1 + sin A) / 2)

Answer

cos(A/2) = sqrt((1 - sin A) / 2)

Answer

cos(A/2) = sin(A/2)

Answer

cos(A/2) = 1 - sin(A/2)

Question 5

How do you prove the trigonometric identity: sin^2(theta) + cos^2(theta) = 1?

Accepted Answer

Use the definitions of sine and cosine, along with the Pythagorean theorem.

Answer

Use the sum-to-product formula.

Answer

Use the double-angle formula.

Answer

Use the half-angle formula.

Question 6

Find the exact value of sin(π/3) using the unit circle.

Accepted Answer

√3/2

Answer

1/2

Answer

√2/2

Answer

0

Question 7

Prove the identity: cos(π/2 - θ) = sin(θ) using the difference formula for cosine.

Accepted Answer

Using the difference formula for cosine

Answer

Using the Pythagorean identity

Answer

Using the cofunction identity

Answer

Using the double-angle formula for cosine

Question 8

Find the domain of the function f(x) = tan(x).

Accepted Answer

All real numbers except multiples of π/2

Answer

All real numbers except multiples of π

Answer

All real numbers

Answer

The function has no domain.

Question 9

Prove the identity: cos(A + B)cos(A - B) = cos²A - sin²B. Use the sum-to-product formula for cosine.

Accepted Answer

Using the sum-to-product formula for cosine

Answer

Using the difference-to-product formula for cosine

Answer

Using the double-angle formula for cosine

Answer

Using the Pythagorean identity

Question 10

Apply the double-angle formula to simplify sin(2x).

Accepted Answer

2sin(x)cos(x)

Answer

2sin(x)

Answer

sin(x)^2

Question 11

Express cos³ θ - sin³ θ in terms of a single trigonometric function.

Accepted Answer

cos 3θ

Answer

cos² θ - sin² θ

Answer

sin 3θ

Answer

sin θ cos θ

Question 12

Use the half-angle identity to express sin(θ/2) in terms of cos θ.

Accepted Answer

√((1 - cos θ)/2)

Answer

√((1 + cos θ)/2)

Answer

sin θ/2

Question 13

Which of the following is the Pythagorean identity?

Accepted Answer

sin²(θ) + cos²(θ) = 1

Answer

tan²(θ) + 1 = sec²(θ)

Answer

sin(θ) / cos(θ) = tan(θ)

Answer

cos(θ) / sin(θ) = cot(θ)

Question 14

What is the double-angle formula for sine?

Accepted Answer

sin(2θ) = 2sin(θ)cos(θ)

Answer

sin(2θ) = sin²(θ) + cos²(θ)

Answer

sin(2θ) = tan(θ) / cot(θ)

Answer

sin(2θ) = sec(θ) - csc(θ)

Question 15

What is the quotient identity for tangent?

Accepted Answer

tan(θ) = sin(θ) / cos(θ)

Answer

tan(θ) = cos(θ) / sin(θ)

Answer

tan(θ) = sec(θ) / csc(θ)

Answer

tan(θ) = cot(θ) / sec(θ)

Question 16

What is the product-to-sum formula for cosine?

Accepted Answer

cos(α)cos(β) = (1/2)[cos(α + β) + cos(α - β)]

Answer

cos(α)cos(β) = (1/2)[cos(α + β) - cos(α - β)]

Answer

cos(α)cos(β) = (1/2)[sin(α + β) + sin(α - β)]

Answer

cos(α)cos(β) = (1/2)[sin(α + β) - sin(α - β)]

Question 17

What is the equation of the unit circle?

Accepted Answer

x² + y² = 1

Answer

(x - h)² + (y - k)² = r²

Answer

y = mx + c

Answer

ax² + bxy + cy² + dx + ey + f = 0

Question 18

Which of the following is the difference formula for cosine?

Accepted Answer

cos(α - β) = cos(α)cos(β) - sin(α)sin(β)

Answer

cos(α - β) = cos(α)cos(β) + sin(α)sin(β)

Answer

cos(α - β) = sin(α)cos(β) - cos(α)sin(β)

Answer

cos(α - β) = sin(α)cos(β) + cos(α)sin(β)

Question 19

What is the double-angle formula for cosine?

Accepted Answer

cos(2θ) = 2cos²(θ) - 1

Answer

cos(2θ) = 1 - 2sin²(θ)

Answer

cos(2θ) = sin²(θ) - cos²(θ)

Answer

cos(2θ) = tan²(θ) - cot²(θ)

Question 20

What is the Pythagorean identity?

Accepted Answer

sin^2(x) + cos^2(x) = 1

Answer

cos(x + y) = cos(x)cos(y) - sin(x)sin(y)

Answer

sin(x + y) = sin(x)cos(y) + cos(x)sin(y)

Answer

tan(x + y) = (tan(x) + tan(y))/1 - tan(x)tan(y)

Question 21

How do you prove the identity: sin^2(x) - cos^2(x) = 1 - 2cos^2(x)?

Accepted Answer

Use the Pythagorean identity and algebraic manipulation.

Answer

Use the double-angle formula for sine.

Answer

Use the half-angle formula for cosine.

Answer

Use the sum-to-product formula for cosine.

Question 22

Find the exact value of sin(π/6) using trigonometric identities.

Accepted Answer

sqrt(3)/2

Answer

1/2

Answer

sqrt(2)/2

Answer

1

Question 23

Simplify the expression: 2cos(x-π/3)cos(x+π/3).

Accepted Answer

1

Answer

cos(2x)

Answer

cos(x)

Answer

sin(2x)

Question 24

Prove the identity: sec(x) - cos(x) = sin(x)tan(x).

Accepted Answer

Use the Pythagorean identity and algebraic manipulation.

Answer

Use the double-angle formula for cosine.

Answer

Use the half-angle formula for tangent.

Answer

Use the product-to-sum formula for sine.

Question 25

Find the period of the function: y = 2 sin (3x + pi / 2)

Accepted Answer

2pi / 3

Answer

pi / 3

Answer

pi

Answer

2pi

Question 26

Find the amplitude of the function: y = 3 cos (2x + pi)

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 27

Which of the following expressions represents the sum formula for sine?

Accepted Answer

sin (x + y) = cos x cos y + sin x sin y

Answer

sin (x + y) = sin x + sin y

Answer

sin (x + y) = sin x - sin y

Answer

sin (x + y) = cos x cos y - sin x sin y

Question 28

Prove the trigonometric identity: $sin(A - B) = sinAcosB - cosAsinB.$ Provide a step-by-step explanation of your proof.

Accepted Answer

$sinAcosB - cosAsinB$

Answer

$cosAcosB + sinAsinB$

Question 29

Which trigonometric identity represents the double angle formula for sine?

Accepted Answer

sin(2θ) = 2sin(θ)cos(θ)

Answer

sin(2θ) = sin(θ) + sin(θ)

Answer

sin(2θ) = cos(2θ)

Answer

sin(2θ) = 2cos(θ)

Question 30

Which trigonometric identity is used to find the cosine of an angle if the sine is known?

Accepted Answer

cos²θ = 1 - sin²θ

Answer

sin²θ + cos²θ = 1

Answer

cscθ = 1 / sinθ

Answer

tanθ = cosθ / sinθ

Question 31

What is the half-angle identity for sine?

Accepted Answer

sin(θ/2) = ±√((1 - cosθ) / 2)

Answer

cos(θ/2) = ±√((1 + cosθ) / 2)

Answer

cot(θ/2) = ±√((1 + cosθ) / (1 - cosθ))

Answer

tan(θ/2) = ±√((1 - sinθ) / (1 + sinθ))

Question 32

Which trigonometric identity is valid for all values of θ?

Accepted Answer

sin²θ + cos²θ = 1

Answer

secθ = 1 / cosθ

Answer

tanθ = sinθ / cosθ

Answer

cotθ = cosθ / sinθ

Question 33

What is the double-angle identity for tangent?

Accepted Answer

tan(2θ) = (2tanθ) / (1 - tan²θ)

Answer

cot(2θ) = (2cotθ) / (1 + cot²θ)

Answer

cos(2θ) = 2cosθ - 1

Answer

sin(2θ) = 2sinθ

Question 34

Which trigonometric identity is used to prove other trigonometric identities?

Accepted Answer

tanθ = sinθ / cosθ

Answer

cotθ = tanθ / sinθ

Answer

secθ = cosθ / sinθ

Answer

sinθ = cosθ / tanθ

Question 35

What is the half-angle identity for tangent?

Accepted Answer

tan(θ/2) = ±√((1 - cosθ) / (1 + cosθ))

Answer

cos(θ/2) = ±√((1 - cosθ) / 2)

Answer

sin(θ/2) = ±√((1 + sinθ) / 2)

Answer

cot(θ/2) = ±√((1 + cosθ) / (1 - cosθ))

Question 36

Which trigonometric identity is utilized specifically to determine the sine of an angle's supplement?

Accepted Answer

sin(π - θ) = sin θ

Answer

csc(θ) = 1/sin θ

Answer

tan(π/2 - θ) = cot θ

Answer

cos(π + θ) = -cos θ

Question 37

Identify the trigonometric identity that simplifies cos² θ + sin² θ to 1.

Accepted Answer

Pythagorean Identity

Answer

Half-Angle Identity

Answer

Sum-to-Product Identity

Answer

Double Angle Identity

Question 38

Which identity is employed to express tan θ in terms of sin θ and cos θ?

Accepted Answer

tan θ = sin θ / cos θ

Answer

sin θ = tan θ / cos θ

Answer

cot θ = sin θ / cos θ

Answer

cos θ = tan θ / sin θ

Question 39

Identify the trigonometric identity utilized to determine the cosine of an angle's complement.

Accepted Answer

cos(π/2 - θ) = sin θ

Answer

csc(θ) = 1/cos θ

Answer

tan(π/2 - θ) = cot θ

Answer

sin(π - θ) = cos θ

Question 40

Which identity is used to represent csc θ in terms of sin θ?

Accepted Answer

csc θ = 1/sin θ

Answer

sec θ = 1/cos θ

Answer

sin θ = 1/csc θ

Answer

cot θ = 1/tan θ

Question 41

Which of the following represents the correct double angle identity for cosine?

Accepted Answer

cos 2θ = cos² θ - sin² θ

Answer

cos 2θ = 1 - 2sin² θ

Answer

cos 2θ = 2cos θ - 1

Answer

cos 2θ = 2sin θ - 1

Question 42

Identify the identity employed to transform sin(α + β) into a product of sines and cosines.

Accepted Answer

Sum-to-Product Identity

Answer

Half-Angle Identity

Answer

Double Angle Identity

Answer

Product-to-Sum Identity

Question 43

Which of the following is the half-angle identity for sine?

Accepted Answer

sin (θ/2) = ±√((1 - cos θ)/2)

Answer

sin (θ/2) = √(2)cos θ

Answer

sin (θ/2) = 1/√(2)sin θ

Answer

sin (θ/2) = √((1 + cos θ)/2)

Question 44

If sinθ = 3/5 and θ is in Quadrant II, what is the value of cosθ?

Accepted Answer

-4/5

Answer

-3/5

Answer

5/4

Answer

4/5

Question 45

Express cos(2θ) in terms of cosθ only.

Accepted Answer

2cos²θ - 1

Answer

2sinθcosθ

Answer

1 - 2sin²θ

Answer

cos²θ - sin²θ

Question 46

Which of the following is the Pythagorean identity?

Accepted Answer

sin²θ + cos²θ = 1

Answer

tan²θ = sec²θ - 1

Answer

sin²θ = cos²θ

Answer

cos²θ = 1 - tan²θ

Question 47

Which of the following is NOT a trigonometric substitution?

Accepted Answer

u = x²

Answer

u = 1 + cos²θ

Answer

u = sin²θ

Answer

u = tanθ

Question 48

Find the integral of ∫cos²x dx.

Accepted Answer

x/2 + sinx/2 + C

Answer

cos²x + C

Answer

sin²x/2 + C

Answer

1/2cos²x + C

Question 49

Which of the following is the half-angle identity for cosine?

Accepted Answer

cos(θ/2) = ±√((1 + cosθ) / 2)

Answer

cos(θ/2) = sin(θ/2)

Answer

cos(θ/2) = tan(θ/2)

Answer

cos(θ/2) = √((1 - cosθ) / 2)

Question 50

Which of the following equations represents the Pythagorean identity?

Accepted Answer

sin²θ + cos²θ = 1

Answer

cot²θ + tan²θ = 1

Answer

sin²θ - cos²θ = 1

Answer

cos²θ - sin²θ = 1

Question 51

What is the double-angle identity for cosine, expressed in terms of sine and cosine?

Accepted Answer

cos(2θ) = cos²θ - sin²θ

Answer

cos(2θ) = 2sin²θ

Answer

cos(2θ) = 2cos²θ

Question 52

Which trigonometric identity helps determine the value of sin(π/2 - θ) in terms of another trigonometric function?

Accepted Answer

sin(π/2 - θ) = cos θ

Answer

sin(π/2 - θ) = tan θ

Answer

sin(π/2 - θ) = cos(π/2 - θ)

Answer

sin(π/2 - θ) = sin θ

Question 53

What is the sum-to-product identity for cosine, expressing the sum of two cosines as a product?

Accepted Answer

cos α + cos β = 2cos((α + β)/2)cos((α - β)/2)

Answer

cos α + cos β = 2sin((α + β)/2)sin((α - β)/2)

Answer

cos α + cos β = 2cos((α - β)/2)cos((α + β)/2)

Answer

cos α + cos β = 2sin((α + β)/2)cos((α - β)/2)

Question 54

Which of the following expressions is NOT a valid trigonometric identity?

Accepted Answer

sin θ = tan θ

Answer

cot θ = 1/tan θ

Answer

cos²θ = 1 - sin²θ

Answer

tan²θ + 1 = sec²θ

Question 55

What is the half-angle identity for tangent, expressing tan(θ/2) in terms of cos θ?

Accepted Answer

tan(θ/2) = ±√((1 - cos θ)/(1 + cos θ))

Answer

tan(θ/2) = ±√((1 + cos θ)/(1 - cos θ))

Answer

tan(θ/2) = √((1 + cos θ)/(1 - cos θ))

Answer

tan(θ/2) = √((1 - cos θ)/(1 + cos θ))

Question 56

Which identity can be used to express (cos(2θ) + 1)/2 in terms of another trigonometric function?

Accepted Answer

(cos(2θ) + 1)/2 = cos²θ

Answer

(cos(2θ) + 1)/2 = cot²θ

Answer

(cos(2θ) + 1)/2 = tan²θ

Answer

(cos(2θ) + 1)/2 = sin²θ

Question 57

What is the triple-angle identity for cosine, expressing cos(3θ) in terms of cos θ?

Accepted Answer

cos(3θ) = 4cos³θ - 3cos θ

Answer

cos(3θ) = 4sin³θ - 3sin θ

Answer

cos(3θ) = 4cos²θ - 3sin θ

Question 58

Which of the following equations represents the product-to-sum identity for sine, expressing the product of two sines as a sum of cosines?

Accepted Answer

sin α sin β = (cos(α - β) - cos(α + β))/2

Answer

sin α sin β = (sin(α - β) - sin(α + β))/2

Answer

sin α sin β = (sin(α - β) + sin(α + β))/2

Answer

sin α sin β = (cos(α - β) + cos(α + β))/2

Question 59

What is the identity for tan(x - y), expressing the tangent of the difference of two angles in terms of the tangents of those angles?

Accepted Answer

tan(x - y) = (tan x - tan y)/(1 + tan x tan y)

Answer

tan(x - y) = (tan x + tan y)/(1 - tan x tan y)

Answer

tan(x - y) = (tan x tan y)/(1 + tan x tan y)