Question

If `( cos theta + sin theta) = sqrt(2) sin theta , " prove that " ( sin theta - cos theta ) = sqrt(2) cos theta`

Answer 1

Given : `cos theta + sin theta  = sqrt(2) sin theta`

We have `( sin theta + cos theta )^2 + (sin theta - cos theta )^2 =2(sin^2 theta + cos^2 theta )`

 `= > ( sqrt(2) sin theta )^2 + ( sin theta - cos theta ) ^2 = 2 `

`= > 2 sin^2 theta + ( sin theta - cos theta ) ^2 = 2`

`= > ( sin theta - cos theta ) ^2 = 2-2 sin^2 theta `

`= > ( sin theta - cos theta ) ^2 =2(1- sin^2 theta)`

`= > ( sin theta - cos theta ) ^2 = 2 cos^2 theta`

`= > ( sin theta - cos theta )  = sqrt(2) cos theta`

Hence proved.