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If cos theta + sin theta = sqrt2 cos the...

If `cos theta + sin theta = sqrt2 cos theta, ` then ` cos theta - sin theta =`

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Given that , `costheta+sintheta=sqrt(2)costheta`
` rArrsintheta=sqrt(2)costheta-costheta`
`rArrsintheta=(sqrt(2)-1)costheta`
Multiply both sides by `(sqrt(2)+1)`
`(sqrt(2)+1)sintheta=(sqrt(2)+1)(sqrt(2)-1)costheta`
`rArrsqrt(2)sintheta+sintheta=costheta`
`sqrt(2)sintheta=costheta-sintheta`
Alternative Method :
We have,
`costheta+sintheta=sqrt(2)costheta`
Squaring both sides , we get
`(costheta+sintheta)^(2)=(sqrt2costheta)^(2)`
`rArrcos^(2)theta+sin^(2)theta+2costhetasintheta=2cos^(2)theta`
`rArr2cos^(2)theta-cos^(2)theta-sin^(2)theta=2sinthetacostheta`
`rArr cos^(2)theta-sin^(2)theta=2sinthetacostheta`
`rArr(costheta-sintheta)(costheta+sintheta)=2sinthetacostheta`
`rArr(costheta-sintheta)*sqrt(2)costheta=2sinthetacostheta`
`:. costheta-sintheta=sqrt(2)sintheta`
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