Question

If cos + sin = 2 cos, prove that cos - sin = +2 sin ​

Answer 1

Answer:

Given

cos θ + sin θ = √2cos θ

Squaring both side, we get

(cos θ + sin θ)2 = 2cos2θ

cos2θ + sin2θ + 2 × cosθ × sinθ = 2cos2θ

sin2θ + 2 × cosθ × sinθ = 2cos2θ – cos2θ

sin2θ + 2 × cosθ × sinθ = cos2θ

cos2θ – 2 × cosθ × sinθ = sin2θ

Now adding sin2θ both side, we get

cos2θ -2 × cosθ × sinθ + sin2θ = sin2θ + sin2θ

(cos θ – sin θ)2 = 2sin2θ

cos θ – sin θ = √2sinθ

∴ cos θ – sin θ = √2sinθ

Answer 2

Answer:

√2sin∅

Step-by-step explanation:

Given

cos θ + sin θ = √2cos θ

Squaring both side, we get

(cos θ + sin θ)2 = 2cos2θ

cos2θ + sin2θ + 2 × cosθ × sinθ = 2cos2θ

sin2θ + 2 × cosθ × sinθ = 2cos2θ – cos2θ

sin2θ + 2 × cosθ × sinθ = cos2θ

cos2θ – 2 × cosθ × sinθ = sin2θ

Now adding sin2θ both side, we get

cos2θ -2 × cosθ × sinθ + sin2θ = sin2θ + sin2θ

(cos θ – sin θ)2 = 2sin2θ

cos θ – sin θ = √2sinθ

∴ cos θ – sin θ = √2sinθ