5.8.1 Proofs involving trigonometric functions Quiz: Pearson Edexcel Maths, Unit 5

The 20 questions

1. Which fundamental trigonometric identity links sine and cosine for any angle theta?

   • sin^2(theta) + cos^2(theta) = 1
   • sin^2(theta) - cos^2(theta) = 1
   • sin(theta) + cos(theta) = 1
   • tan^2(theta) + 1 = sec^2(theta)

2. What is the equivalent expression for tan(theta) in terms of sine and cosine?

   • sin(theta) / cos(theta)
   • 1 / cos(theta)
   • 1 / sin(theta)
   • cos(theta) / sin(theta)

3. What is the reciprocal identity for secant in terms of cosine?

   • 1 / cos(theta)
   • 1 / tan(theta)
   • 1 / sin(theta)
   • cos(theta) / sin(theta)

4. What is the reciprocal identity for cosecant in terms of sine?

   • 1 / tan(theta)
   • 1 / sin(theta)
   • sin(theta) / cos(theta)
   • 1 / cos(theta)

5. What is the reciprocal identity for cotangent in terms of tangent?

   • 1 / tan(theta)
   • 1 / cos(theta)
   • tan(theta) / sin(theta)
   • 1 / sin(theta)

6. Which identity relates the tangent function directly to the secant function?

   • tan^2(theta) - 1 = sec^2(theta)
   • 1 + tan^2(theta) = sec^2(theta)
   • sin^2(theta) + cos^2(theta) = 1
   • 1 + cot^2(theta) = cosec^2(theta)

7. Which identity relates the cotangent function directly to the cosecant function?

   • sin^2(theta) + cos^2(theta) = 1
   • cot^2(theta) - 1 = cosec^2(theta)
   • 1 + cot^2(theta) = cosec^2(theta)
   • 1 + tan^2(theta) = sec^2(theta)

8. What is the derivative of sin(x) used often in trigonometric proof steps?

   • sin(x)
   • -sin(x)
   • cos(x)
   • -cos(x)

9. Simplify the trigonometric expression (1 - cos^2(theta)) using standard identities.

   • cos^2(theta)
   • sin^2(theta)
   • tan^2(theta)
   • sec^2(theta)

10. Simplify the expression (sec^2(theta) - 1) using standard trigonometric identities.

    • cot^2(theta)
    • tan^2(theta)
    • sin^2(theta)
    • cosec^2(theta)

11. Simplify the expression (cosec^2(theta) - 1) using standard trigonometric identities.

    • tan^2(theta)
    • sec^2(theta)
    • cos^2(theta)
    • cot^2(theta)

12. Evaluate the exact value of sin^2(45 degrees) + cos^2(45 degrees).

    • 2
    • 0
    • 1
    • 0.5

13. Rewrite the fraction sin(theta) / cos^2(theta) using secant and tangent.

    • sin(theta)sec(theta)
    • tan(theta)cosec(theta)
    • cot(theta)cosec(theta)
    • tan(theta)sec(theta)

14. Express tan^2(theta) solely in terms of cosine using identities.

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

15. What algebraic method proves sec^4 theta - tan^4 theta equals sec^2 theta + tan^2 theta?

    • Completing the square
    • Polynomial long division
    • Integration by parts
    • Difference of two squares

16. When proving a trigonometric identity, which side should you generally start with?

    • More complex side
    • Simpler side
    • Right-hand side only
    • Left-hand side only

17. Prove cos^4(theta) - sin^4(theta) is identically equal to which simpler expression?

    • 1 - 2sin^2(theta)
    • cos^2(theta) - sin^2(theta)
    • cos^2(theta) + sin^2(theta)
    • 2cos^2(theta) - 1

18. Find the range of values for k if sec(theta) = k - 2 has no real solutions.

    • -1 < k < 3
    • k >= 3
    • 1 < k < 3
    • k > 3 or k < -1

19. Prove that (tan(theta) + cot(theta)) is identically equal to which product expression?

    • sec^2(theta)
    • cosec^2(theta)
    • sin(theta)cos(theta)
    • sec(theta)cosec(theta)

20. What is the result of expanding (sin(theta) + cos(theta))^2 using trigonometric identities?

    • 1 - 2sin(theta)cos(theta)
    • 1 + 2sin(theta)cos(theta)
    • 1 + sin(theta)cos(theta)
    • 2 + sin(theta)cos(theta)

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