1.

If sin θ + cos θ = 1, then the value of cos 2θ is:1. ±12. 03. \(\frac{\sqrt3}{2}\)4. \(\frac1{2+\sqrt2}\)

Answer» Correct Answer - Option 1 : ±1

Concept:

Trigonometric Identities:

  • sin2 θ + cos2 θ = 1.
  • sin 2θ = 2 sin θ cos θ.
  • cos 2θ = cos2 θ - sin2 θ.

 

Calculation:

Given: sin θ + cos θ = 1

Squaring both sides, we get:

⇒ sin2 θ + cos2 θ + 2 sin θ cos θ = 1

⇒ 1 + sin 2θ = 1

⇒ sin 2θ = 0

Using sin2 θ + cos2 θ = 1, we can say:

sin2 2θ + cos2 2θ = 1

⇒ 0 + cos2 2θ = 1

⇒ cos 2θ = ±1.

 

Trigonometric Identities:

  • sin (A ± B) = sin A cos B ± sin B cos A.
  • cos (A ± B) = cos A cos B ∓ sin A sin B.


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