Question

if cos theta + sin theta =1 then prove that cos theta-sin Theta= + - 1 please solve this

Answer 1

cosß+sinß=1
or, (cosß+sinß)^2 = 1
or, cosß^2+sinß^2 + 2cosßsinß= 1
or, cosßsinß=0

Then, (Cosß - sinß)^2 = cosß^2+sinß^2 - 2cosßsinß
= 1 - 0
= 1
Cosß - sinß = +-1 (proved)

Answer 2

Answer :

Given that, cosθ + sinθ = 1

⇒ (cosθ + sinθ)² = 1²

⇒ cos²θ + sin²θ + 2 sinθ cosθ = 1

⇒ 1 + 2 sinθ cosθ = 1

⇒ 2 sinθ cosθ = 0

⇒ sinθ cosθ = 0

Now, (cosθ - sinθ)²

= (cosθ + sinθ)² - 4 sinθ cosθ

= 1² - (4 × 0)

= 1 - 0

= 1

⇒ (cosθ - sinθ)² = 1

⇒ cosθ - sinθ = +_ 1

Hence, proved.

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