Question

Q] If cos θ + sin θ = √2cos θ , then prove that
cos θ - sin θ = √2sin θ

•No spam.
​

Answer 1

$\huge\color{pink}\boxed{\colorbox{Black}{❥ Answer}}$

Given that: cos x + sin x = √2 cos x, or

sin x = cos x (√2–1)

Next to prove, cos x - sin x = √2 sin x, take

LHS = cos x - sin x

= [sin x *1/(√2–1)]- sin x

= sin x*[1-(√2–1)]/(√2–1)

= sin x [2-√2]*(√2+1)/(√2–1)(√2+1)

= sin x* [2√2+2–2-√2]

= sin x*√2 = RHS. Proved

Answer 2

VeriFied√

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

@Haryanvi_chankya