Question

Let cos theta+ sin theta=root 2 cos theta.show that cos theta-sin theta=root2 sin theta

Answer 1

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If sinϴ+cosϴ = √2 cosϴ prove that cosϴ-sinϴ = √2 sinϴ,  (ϴ is an acute angle)

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Given `sin theta + cos theta = sqrt(2)cos theta` prove `cos theta - sin theta = sqrt(2)sin theta`

`sin theta + cos theta = sqrt(2)cos theta` square both sides

`sin^2theta + cos^2 theta + 2sintheta costheta=2cos^2theta`

`sin^2theta - cos^2theta + 2sinthetacostheta=0`

`-sin^2theta + cos^2theta -2sinthetacostheta=0` Add `2sin^2theta` to both sides

`sin^2theta+cos^2theta-2sinthetacostheta=2sin^2theta`

`(costheta-sintheta)^2=2sin^2theta`

`costheta-sintheta=sqrt(2)sintheta` as required.