Tough Trigonometric Identities Test - 23+ MCQs: Trigonometric Identities Quiz: Test Your Understanding with Essential Identity Questions and Answers

1. What is the half-angle identity for sine?

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

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

sin(theta/2) = sqrt(1 - sin(theta))/2

sin(theta/2) = sqrt(1 + sin(theta))/2

2. What is the sum-to-product identity for secant and cosecant?

sec(theta) + sec(phi) = 2sec((theta + phi)/2)cos((theta - phi)/2)

csc(theta) + csc(phi) = 2csc((theta + phi)/2)sin((theta - phi)/2)

sec(theta) + csc(phi) = sec(theta + phi) + sec(theta - phi)

csc(theta) + sec(phi) = csc(theta + phi) - csc(theta - phi)

3. What is the co-function identity for sine?

sin(pi/2 - theta) = cos(theta)

sin(2theta) = 2sin(theta)cos(theta)

sin(theta) = tan(theta)/cot(theta)

sin(theta) = 1/csc(theta)

4. What is the sum-to-product identity for tangent?

tan(theta) + tan(phi) = sin(theta + phi)/cos(theta + phi)

tan(theta) + tan(phi) = tan((theta + phi)/2)/1 - tan((theta + phi)/2)

tan(theta) + tan(phi) = sin(theta + phi)/cos(theta - phi)

tan(theta) + tan(phi) = cot((theta + phi)/2)/1 + cot((theta + phi)/2)

5. What is the reciprocal identity for cosine?

csc(theta) = 1/sin(theta)

sec(theta) = 1/cos(theta)

cot(theta) = 1/tan(theta)

cos(theta) = 1/sin(theta)

6. Prove the identity: cos²(x) - sin²(x) = 1

Use the Pythagorean identity

Apply double-angle formulas

Use the quotient identity

Apply the sum-to-product formula

7. What is the difference-to-product identity for sine and cosine?

sin(theta) - sin(phi) = 2cos((theta + phi)/2)sin((theta - phi)/2)

cos(theta) - cos(phi) = 2sin((theta + phi)/2)sin((theta - phi)/2)

sin(theta) - cos(phi) = sin(theta + phi) - sin(theta - phi)

cos(theta) - sin(phi) = cos(theta + phi) + cos(theta - phi)

8. Which identity is true for all values of theta?

tan(theta) = sin(theta)/cos(theta)

cot(theta) = cos(theta)/sin(theta)

sin(2theta) = 2sin(theta)cos(theta)

csc^2(theta) - cot^2(theta) = 1

9. Which identity involves both sine and cosine?

tan(theta) = sin(theta)/cos(theta)

cot(theta) = cos(theta)/sin(theta)

sin(2theta) = 2sin(theta)cos(theta)

cot(2theta) = cot^2(theta) - 1

10. What is the Pythagorean Identity for tangent?

tan^2(theta) = 1 + cot^2(theta)

tan^2(theta) + 1 = sec^2(theta)

tan^2(theta) + 1 = csc^2(theta)

cot^2(theta) = sec^2(theta) - 1

11. What is the half-angle identity for cosine?

cos(theta/2) = sqrt((1 + cos(theta))/2)

cos(theta/2) = sqrt((1 - cos(theta))/2)

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

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

12. What is the product-to-sum identity for sine?

sin(theta)sin(phi) = 1/2[cos(theta - phi) - cos(theta + phi)]

sin(theta)sin(phi) = 1/2[cos(theta + phi) + cos(theta - phi)]

sin(theta)sin(phi) = 1/2[sin(theta + phi) - sin(theta - phi)]

sin(theta)sin(phi) = 1/2[sin(theta - phi) + sin(theta + phi)]

13. Find the exact value of: sin(2x) given that sin(x) = 3/5

24/25

4/5

-24/25

-4/5

14. What is the Pythagorean Identity?

sin^2(theta) = 1 - cos^2(theta)

tan(theta) = sin(theta)/cos(theta)

cot(theta) = cos(theta)/sin(theta)

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

15. What is the quotient identity for tangent?

tan(theta) = sin(theta)/cos(theta)

cot(theta) = cos(theta)/sin(theta)

tan(theta) = 1/cot(theta)

tan(theta) = cos(theta)/sin(theta)

16. What is the sum-to-product identity for cotangent?

cot(theta)cot(phi) = tan((theta + phi)/2)/1 - tan((theta + phi)/2)

cot(theta)cot(phi) = cot((theta + phi)/2)/1 + cot((theta + phi)/2)

cot(theta)cot(phi) = sin(theta + phi)/cos(theta + phi)

cot(theta)cot(phi) = 1/2[cot(theta + phi) + cot(theta - phi)]

17. Which identity is known as the Power-Reducing or Half-Angle identity?

cos^2(theta) = (1 + cos(2theta))/2

sin^2(theta) = (1 - cos(2theta))/2

tan^2(theta) = (1 - cos(2theta))/(1 + cos(2theta))

cot^2(theta) = (1 + cos(2theta))/sin^2(theta)

18. What is the sum-to-product identity for sine and cosine?

sin(theta) + sin(phi) = 2sin((theta + phi)/2)cos((theta - phi)/2)

cos(theta) + cos(phi) = 2cos((theta + phi)/2)sin((theta - phi)/2)

sin(theta) + cos(phi) = sin(theta + phi) + sin(theta - phi)

cos(theta) + sin(phi) = cos(theta + phi) - cos(theta - phi)

19. Simplify the expression: tan(x) / cot(x)

sin(x)

cos(x)

sec(x)

csc(x)

20. What is the double-angle identity for cosine?

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

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

cos(2theta) = 1 - 2sin^2(theta)

cos(2theta) = 2sin^2(theta) - 1

Trigonometric Identities MCQ Test 1