Trigonometric Identities: Master the Building Blocks of Trigonometry

Question 1

Which of the following is the double angle identity for the sine of 2x?

Accepted Answer

2 sin x cos x

Answer

2 sin x

Answer

sin^2 x

Answer

1 - 2 sin^2 x

Question 2

Identify the option that is NOT a trigonometric identity:

Accepted Answer

sin x = x

Answer

cos (2x) = 2 cos^2 x - 1

Answer

tan x = sin x / cos x

Answer

sin^2 x + cos^2 x = 1

Question 3

Given that sin x = 0.5, calculate the value of cos x:

Accepted Answer

√3 / 2

Answer

0.5

Answer

-√3 / 2

Answer

1

Question 4

Simplify the following expression: 2 sin x cos x

Accepted Answer

sin 2x

Answer

2 cos x

Answer

2 sin x

Answer

cos 2x

Question 5

Which of the following is the Pythagorean identity?

Accepted Answer

sin^2 x + cos^2 x = 1

Answer

tan^2 x + 1 = sec^2 x

Answer

cot^2 x - 1 = cosec^2 x

Answer

sin x + cos x = 1

Question 6

Identify the half-angle identity from the following options:

Accepted Answer

sin (x/2) = ±√((1 - cos x) / 2)

Answer

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

Answer

tan x = sin x / cos x

Question 7

Solve the equation: sin (2x - π/3) = 0

Accepted Answer

x = π/6, 5π/6

Answer

x = π/2, 3π/2

Answer

x = π/3, 2π/3

Answer

x = 0, π

Question 8

Which of the following is the fundamental trigonometric identity that relates the sine and cosine functions?

Accepted Answer

sin²(θ) + cos²(θ) = 1

Answer

tan²(θ) - cot²(θ) = 1

Answer

tan²(θ) + cot²(θ) = 1

Answer

sin²(θ) - cos²(θ) = 1

Question 9

What is the trigonometric ratio of the complementary angle of θ?

Accepted Answer

sin(90° - θ) = cos(θ)

Answer

sin(90° - θ) = sin(θ)

Answer

cos(90° - θ) = sin(θ)

Question 10

What is the half-angle identity for sine?

Accepted Answer

sin(θ/2) = ±√((1 - cos(θ))/2)

Answer

sin(θ/2) = √((1 - sin(θ))/2)

Answer

sin(θ/2) = √((1 + cos(θ))/2)

Answer

sin(θ/2) = ±√((1 + sin(θ))/2)

Question 11

What is the result of applying the double-angle identity for tangent to tan(2θ)?

Accepted Answer

tan(2θ) = (2tan(θ))/(1 - tan²(θ))

Answer

tan(2θ) = (1 + tan²(θ))/(2tan(θ))

Answer

tan(2θ) = (2tan(θ))/(tan²(θ) - 1)

Question 12

Which trigonometric identity is commonly used as a fundamental building block for proving other trigonometric identities?

Accepted Answer

sin²(θ) + cos²(θ) = 1

Answer

sin(θ)cos(θ) = (1/2)sin(2θ)

Answer

tan(θ) = sin(θ)/cos(θ)

Question 13

Simplify the expression (sin(θ) + cos(θ))².

Accepted Answer

1 + sin(2θ)

Answer

2cos(θ) - 2sin(θ)

Answer

1 - sin(2θ)

Answer

2sin(θ)cos(θ)

Question 14

Which of the following trigonometric identities is known as the Pythagorean identity?

Accepted Answer

sin²θ + cos²θ = 1

Answer

tan²θ + sec²θ = 1

Answer

sinθ + cosθ = 1

Answer

tanθ + secθ = 1

Question 15

Simplify the trigonometric expression: cos(2θ)

Accepted Answer

cos²θ - sin²θ

Answer

sin²θ - cos²θ

Answer

2sin²θ

Answer

2cos²θ

Question 16

Identify the correct trigonometric identity:

Accepted Answer

sin²x + cos²x = 1

Answer

sinx + cosx = 0

Answer

cosecx + secx = 2

Answer

tanx + cotx = 1

Question 17

Simplify the trigonometric expression: 2sinθcosθ

Accepted Answer

sin2θ

Answer

tan2θ

Answer

1

Answer

cos2θ

Question 18

Find the value of tan(π/3).

Accepted Answer

√3

Answer

2

Answer

0

Answer

1

Question 19

Which of the following is a Pythagorean trigonometric identity?

Accepted Answer

sin²x + cos²x = 1

Answer

tanx + cotx = 1

Answer

sinx + cosx = 0

Answer

cosecx + secx = 2

Question 20

Which trigonometric identity expresses the relationship between sine and cosine for all angles?

Accepted Answer

sin²θ + cos²θ = 1

Answer

tan²θ + 1 = sec²θ

Answer

sin(A - B) = sinAcosB - cosAsinB

Answer

cos(A + B) = cosAcosB - sinAsinB

Question 21

Which double-angle identity expresses the cosine of twice an angle?

Accepted Answer

cos(2θ) = cos²θ - sin²θ

Answer

tan(2θ) = 2tanθ/(1 - tan²θ)

Answer

sin(2θ) = 2sinθcosθ

Answer

cot(2θ) = (1 - cot²θ)/2cotθ

Question 22

What is the half-angle identity for tangent?

Accepted Answer

tan(θ/2) = ±√((1 - cosθ)/(1 + cosθ))

Answer

sin(θ/2) = ±√((1 - sinθ)/(1 + sinθ))

Answer

cos(θ/2) = ±√((1 + cosθ)/(1 - cosθ))

Answer

cot(θ/2) = ±√((1 + cotθ)/(1 - cotθ))

Question 23

Find the exact value of cos(15° - 45°).

Accepted Answer

(√6 - √2)/4

Answer

(√6 - √2)/2

Answer

(-√6 - √2)/4

Answer

(-√6 - √2)/2

Question 24

Simplify the expression: 4sinθcosθcosθ

Accepted Answer

2sinθcos²θ

Answer

sin2θcosθ

Answer

cos2θsinθ

Answer

2cos²θsinθ

Question 25

To find the sine of half an angle, which identity should be used?

Accepted Answer

sin(θ/2) = ±√((1 - cosθ)/2)

Answer

cos(θ/2) = ±√((1 + cosθ)/2)

Answer

tan(θ/2) = ±√((1 + tanθ)/2)

Answer

cot(θ/2) = ±√((1 - cotθ)/2)

Question 26

Which of the following is the Pythagorean identity?

Accepted Answer

sin²θ + cos²θ = 1

Answer

1 - sin²θ = cos²θ

Answer

tan²θ + cot²θ = 1

Answer

tanθ / cotθ = 1

Question 27

Find the value of sin(90°-θ).

Accepted Answer

cosθ

Answer

cotθ

Answer

sinθ

Answer

tanθ

Question 28

Simplify the expression: 2sinθcosθ using double-angle identities.

Accepted Answer

sin2θ

Answer

cot2θ

Answer

cos2θ

Answer

tan2θ

Question 29

Simplify the expression: (1+tan²θ)/(1+cot²θ).

Accepted Answer

tan²θ

Answer

1

Answer

sin²θ

Answer

cot²θ

Question 30

Find the amplitude of the function y = 3cos(2θ).

Accepted Answer

3

Answer

-3

Answer

6

Answer

1.5

Question 31

Which of the following is the identity used to find the value of sin²θ?

Accepted Answer

sin²θ = (1 - cos2θ) / 2

Answer

tan²θ = sin²θ / cos²θ

Answer

cos²θ = (1 + cos2θ) / 2

Answer

cot²θ = cos²θ / sin²θ

Question 32

Which of the following identities helps us express tan(θ/2) in terms of tanθ?

Accepted Answer

tan(θ/2) = (1 - cosθ) / (1 + cosθ)

Answer

tanθ = sinθ / cosθ

Answer

cot(θ/2) = (1 + cosθ) / (1 - cosθ)

Answer

cotθ = cosθ / sinθ

Question 33

What is the half-angle identity for sin(θ/2)?

Accepted Answer

sin(θ/2) = ±√((1 - cosθ) / 2)

Answer

sin(θ/2) = √(2(1 + cosθ))

Answer

sin(θ/2) = ±√((1 + cosθ) / 2)

Answer

sin(θ/2) = √(2(1 - cosθ))

Question 34

Which trigonometric substitution is used to integrate √(1 - x²)?

Accepted Answer

x = sinθ

Answer

x = cosθ

Answer

x = cotθ

Answer

x = tanθ

Question 35

Simplify the expression: 2cos³θ - 3cosθ

Accepted Answer

cosθ(2cos²θ - 3)

Answer

cosθ(2cosθ + 3)

Answer

sinθ(2sinθ + 3)

Answer

sinθ(2sin²θ - 3)

Question 36

Find the value of sin(15°)cos(75°) - cos(15°)sin(75°)

Accepted Answer

0

Answer

1

Answer

1 / 2

Answer

-1

Question 37

Simplify the expression: (sinθ - cosθ)²

Accepted Answer

1 - 2sinθcosθ

Answer

cos²θ - sin²θ

Answer

1 + 2sinθcosθ

Answer

sin²θ - cos²θ

Question 38

Which of the following expressions is equivalent to sin(x + π/2)?

Accepted Answer

cos(x)

Answer

-cos(x)

Answer

sin(x)

Answer

-sin(x)

Question 39

Simplify the expression: (1 - cos 2θ) / sin 2θ

Accepted Answer

tan θ

Answer

cos θ

Answer

sin θ

Answer

cot θ

Question 40

If sin A = 1/√5 and cos B = 2/√13, where A and B are acute angles, find the value of cos(A + B).

Accepted Answer

(2 - 3√13) / (√65)

Answer

(3 - 2√13) / (√65)

Answer

(2 + 3√13) / (√65)

Answer

(-2 - 3√13) / (√65)

Question 41

Which of the following is the Pythagorean identity?

Accepted Answer

sin²(θ) + cos²(θ) = 1

Answer

sin(θ) + cos(θ) = 1

Answer

tan(θ) ⋅ cot(θ) = 1

Answer

tan²(θ) + cot²(θ) = 1

Question 42

If sin(θ) = 1/2, what is cos(θ)?

Accepted Answer

√3/2

Answer

-√3/2

Answer

√2/2

Answer

1/2

Question 43

Which of the following identities is true for complementary angles?

Accepted Answer

sin(θ) = cos(90° - θ)

Answer

cot(θ) = tan(90° - θ)

Answer

cos(θ) = sin(90° + θ)

Answer

sin(θ) = tan(90° - θ)

Question 44

Simplify the expression: 2 sin(θ)⋅cos(θ)

Accepted Answer

sin(2θ)

Answer

2 sin(θ)

Answer

cos(2θ)

Answer

2 cos(θ)

Question 45

Solve the trigonometric equation: 2 sin²(θ) - 1 = 0

Accepted Answer

θ = π/6, 5π/6

Answer

θ = 0, π

Answer

θ = π/3, 2π/3

Answer

θ = π/4, 3π/4

Question 46

Simplify the expression: cos(π/3)⋅cos(π/4) - sin(π/3)⋅sin(π/4)

Accepted Answer

1/2

Answer

√2/2

Answer

0

Question 47

Which of the following trigonometric identities can be used to derive the double-angle identity for sine?

Accepted Answer

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

Answer

cos(2θ) = 1 - 2sin²(θ)

Answer

sin²(θ) = (1 - cos(2θ))/2

Answer

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

Question 48

Which of the following is the Pythagorean identity?

Accepted Answer

sin²θ + cos²θ = 1

Answer

sec²θ - tan²θ = 1

Answer

tan²θ + cot²θ = 1

Question 49

What is the sine of the complementary angle of 30°?

Accepted Answer

cos 30°

Answer

cosec 30°

Answer

sin 30°

Answer

tan 30°

Question 50

Simplify the expression: 2sin θ cos θ

Accepted Answer

sin 2θ

Answer

cos 2θ

Answer

tan 2θ

Answer

cosec 2θ

Question 51

Find the value of cos² 45° - sin² 45°

Accepted Answer

0

Answer

-1

Answer

1

Answer

1/2

Question 52

Which of the following is NOT a trigonometric identity?

Accepted Answer

sin θ / cos θ = tan θ

Answer

tan θ = sin θ / cos θ

Answer

cosec θ = 1 / sin θ

Answer

cos² θ - sin² θ = 1

Question 53

Simplify the expression: (sin θ + cos θ)²

Accepted Answer

1 + sin 2θ

Answer

1 - sin 2θ

Answer

1 + 2sin θ cos θ

Answer

1 - 2sin θ cos θ

Question 54

Find the value of sin 150°

Accepted Answer

-1/2

Answer

√3/2

Answer

1/2

Answer

-√3/2

Question 55

Which of the following identities is used to prove the Pythagorean identity?

Accepted Answer

(sin θ + cos θ)² + (sin θ - cos θ)² = 2

Answer

(sin θ + cos θ)² - (sin θ - cos θ)² = 2

Answer

(sin θ - cos θ) / (sin θ + cos θ) = 2

Answer

(sin θ + cos θ) / (sin θ - cos θ) = 2

Question 56

Find the value of tan (2π/3 + π/6)

Accepted Answer

-√3

Answer

0

Answer

1

Answer

√3

Question 57

Which of the following is the correct form of the double-angle identity for cosine?

Accepted Answer

cos 2θ = 2cos²θ - 1

Answer

cos 2θ = cos²θ - 2sin²θ

Answer

cos 2θ = 2sin²θ - cos²θ

Answer

cos 2θ = 1 - 2cos²θ