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If cosθ + sinθ = √2 cosθ, show that cosθ – sinθ = √2 sinθ. - Mathematics

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Sum

If cosθ + sinθ = √2 cosθ, show that cosθ – sinθ = √2 sinθ.

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Solution

We have,
cosθ + sinθ = cosθ

`⇒ (cosθ + sinθ)2 = 2 cos^2 θ`

`⇒ cos^2 θ + sin^2 θ + 2 cosθsinθ = 2 cos^2 θ`

`⇒ cos^2 θ – 2cosθ sinθ = sin2θ`

`⇒ cos^2 θ – 2cosθsinθ + sin^2 θ = 2sin^2 θ`

`⇒ (cosθ – sinθ)^2 = 2sin^2 θ`

`⇒ cosθ – sinθ = √2 sinθ`

Concept: Trigonometric Identities
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