Q) If cosθ + sin θ = 1 , then prove that cosθ – sinθ = ±1
Ans:
VIDEO SOLUTION
STEL BY STEP SOLUTION
Given that If cosθ + sinθ = 1
Step 1: We square the above equation on both sides, we get:
(cos θ + sin θ) = (1)
∴ cos2 θ + sin2 θ + 2 cos θ sin θ = 1
∴ cos2 θ + sin2 θ = 1 – 2 cos θ sin θ
Step 2: Now, by Trigonometry identity, we have, cos2 θ + sin2 θ = 1
∴ cos2 θ + sin2 θ = cos2 θ + sin2 θ – 2 cos θ sin θ
Step 3: Now since a2 + b2 – 2 a b = (a – b)2
∴ cos2 θ + sin2 θ = (cos θ – sin θ)2
∴ 1 = (cos θ – sin θ)2 (by cos2 θ + sin2 θ = 1)
Step 4: Now by taking square root on both sides, we get:
√ (1) = √ (cos θ – sin θ)2
∴ cosθ – sinθ = ±1 …… Hence proved!
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