Identity sin^2A + cos^2A = 1 MCQs Quiz | Class 10

This quiz is for **Class X** students studying **Mathematics (Code 041)**, specifically from **Unit V: Trigonometry**. The topic covered is the fundamental trigonometric **Identity sin^2A + cos^2A = 1**, focusing on its proof and direct applications. Test your understanding and solidify your knowledge of this essential identity. After attempting all questions, click “Submit Quiz” to see your score and “Download Answer PDF” for a detailed review.

Understanding the Identity sin^2A + cos^2A = 1

The trigonometric identity sin^2A + cos^2A = 1 is one of the most fundamental and crucial identities in trigonometry. It forms the basis for deriving many other identities and solving various trigonometric problems. This identity holds true for any angle A.

Proof of the Identity

Consider a right-angled triangle PQR, right-angled at Q. Let angle PRQ be A.

  • By definition, sine of angle A (sin A) = Opposite side / Hypotenuse = PQ / PR.
  • By definition, cosine of angle A (cos A) = Adjacent side / Hypotenuse = QR / PR.

From the definitions, we can write:

  • sin^2 A = (PQ / PR)^2 = PQ^2 / PR^2
  • cos^2 A = (QR / PR)^2 = QR^2 / PR^2

Now, let’s add them: sin^2 A + cos^2 A = (PQ^2 / PR^2) + (QR^2 / PR^2) sin^2 A + cos^2 A = (PQ^2 + QR^2) / PR^2

According to the Pythagoras theorem in triangle PQR: PQ^2 + QR^2 = PR^2

Substitute this into our sum: sin^2 A + cos^2 A = PR^2 / PR^2 sin^2 A + cos^2 A = 1

This proves the identity.

Direct Applications

This identity has several direct applications in solving problems and simplifying expressions:

  1. Finding one ratio when the other is known: If you know the value of sin A, you can find cos A (and vice versa) using this identity. Example: If sin A = 3/5, then (3/5)^2 + cos^2 A = 1 => 9/25 + cos^2 A = 1 => cos^2 A = 1 – 9/25 = 16/25 => cos A = 4/5 (assuming A is in the first quadrant).
  2. Simplifying trigonometric expressions: Many complex expressions can be simplified by replacing sin^2 A + cos^2 A with 1, or by replacing 1 – sin^2 A with cos^2 A, or 1 – cos^2 A with sin^2 A. Example: Simplify (1 – sin^2 A) / cos^2 A. Since 1 – sin^2 A = cos^2 A, the expression becomes cos^2 A / cos^2 A = 1.
  3. Deriving other identities: This identity is fundamental to deriving other Pythagorean identities:
    • Dividing by cos^2 A: sin^2 A/cos^2 A + cos^2 A/cos^2 A = 1/cos^2 A => tan^2 A + 1 = sec^2 A
    • Dividing by sin^2 A: sin^2 A/sin^2 A + cos^2 A/sin^2 A = 1/sin^2 A => 1 + cot^2 A = cosec^2 A

Quick Revision Points

  • The identity is: sin^2 A + cos^2 A = 1.
  • It is derived from the Pythagorean theorem in a right-angled triangle.
  • Rearrangements are:
    • sin^2 A = 1 – cos^2 A
    • cos^2 A = 1 – sin^2 A
  • Use it to find sin A if cos A is known, or cos A if sin A is known.
  • It helps simplify trigonometric expressions significantly.
  • It is the basis for tan^2 A + 1 = sec^2 A and 1 + cot^2 A = cosec^2 A.

Practice Questions (Extra)

  1. If sin A = 1/2, what is the value of cos A? (Assume A is acute)
  2. Simplify: sin^2 75 degrees + cos^2 75 degrees.
  3. What is the value of (sec^2 A – tan^2 A)? Hint: Consider dividing sin^2 A + cos^2 A = 1 by cos^2 A.
  4. If cos A = 5/13, find sin A. (Assume A is acute)
  5. Show that (1 – cos A)(1 + cos A) = sin^2 A.

These extra questions will help you further practice and master the applications of the identity sin^2 A + cos^2 A = 1.