Simple Trigonometric Identities MCQs Quiz | Class 10

This quiz covers important Multiple Choice Questions (MCQs) on Class X Mathematics (Code 041), from Unit V: Trigonometry, specifically focusing on Simple Trigonometric Identities. You will practice deriving simple identities using the fundamental sin^2(theta) + cos^2(theta) = 1 and various simplification techniques. Test your knowledge by attempting all 10 questions, then submit to see your score and review answers. Don’t forget to download the PDF for future reference!

Understanding Simple Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variables for which both sides of the equation are defined. These identities are fundamental in trigonometry, simplifying complex expressions, and solving trigonometric equations. Class 10 focuses on the basic identity involving sine and cosine, and its applications.

The Pythagorean Identity

The most fundamental trigonometric identity, derived from the Pythagorean theorem, is:

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

This identity holds true for any angle theta.

Derivation and Related Identities

  • From sin^2(theta) + cos^2(theta) = 1:
    • Divide by cos^2(theta): (sin^2(theta)/cos^2(theta)) + (cos^2(theta)/cos^2(theta)) = 1/cos^2(theta)
      This simplifies to: tan^2(theta) + 1 = sec^2(theta) or 1 + tan^2(theta) = sec^2(theta)
    • Divide by sin^2(theta): (sin^2(theta)/sin^2(theta)) + (cos^2(theta)/sin^2(theta)) = 1/sin^2(theta)
      This simplifies to: 1 + cot^2(theta) = cosec^2(theta)

Summary of Key Identities

Identity Description
sin^2(A) + cos^2(A) = 1 Pythagorean Identity
1 + tan^2(A) = sec^2(A) Derived from Pythagorean Identity
1 + cot^2(A) = cosec^2(A) Derived from Pythagorean Identity
tan(A) = sin(A) / cos(A) Quotient Identity
cot(A) = cos(A) / sin(A) Quotient Identity
sec(A) = 1 / cos(A) Reciprocal Identity
cosec(A) = 1 / sin(A) Reciprocal Identity
cot(A) = 1 / tan(A) Reciprocal Identity

Simplification Techniques

  • Use Fundamental Identities: Look for sin^2 + cos^2 = 1, 1 + tan^2 = sec^2, 1 + cot^2 = cosec^2 directly.
  • Convert to Sine and Cosine: Often, converting all trigonometric ratios (tan, cot, sec, cosec) into their equivalent sine and cosine forms helps simplify expressions. For example, replace tan(A) with sin(A)/cos(A) and sec(A) with 1/cos(A).
  • Factorization: Use algebraic factorization techniques (like a^2 - b^2 = (a-b)(a+b)) with trigonometric terms.
  • Common Denominators: When adding or subtracting fractions, find a common denominator.

Quick Revision

  1. Always remember sin^2(A) + cos^2(A) = 1.
  2. The other two Pythagorean identities (1 + tan^2(A) = sec^2(A) and 1 + cot^2(A) = cosec^2(A)) can be derived from the first one.
  3. Practice converting all ratios to sine and cosine.
  4. Look for algebraic patterns (e.g., (a+b)^2, (a-b)^2, a^2-b^2).

Extra Practice Questions

  1. Simplify sin(A)sec(A).
  2. Prove that (1 - tan^2(A)) / (cot^2(A) - 1) = tan^2(A).
  3. If sec(A) + tan(A) = x, then sec(A) - tan(A) = ?.
  4. Simplify (1 + tan^2(A))cos^2(A).
  5. Prove (sin(theta) + cosec(theta))^2 + (cos(theta) + sec(theta))^2 = 7 + tan^2(theta) + cot^2(theta).