Question

Let t = cosA + sinA
find values of t if
2 - sin2A = 2(cosA + sinA)

Answer 1

`2 - \sin(2A) = 2(\cos A + \sin A)\\\\2-2\sin A\cos A=2\cos A+2\sin A\\\\2\sin A\cos A+2\cos A+2\sin A-2=0\\\\1+2\sin A\cos A+2\cos A+2\sin A-3=0\\\\\sin^2 A+\cos^2 A+2\sin A\cos A+2\cos A+2\sin A-3=0\\\\(\sin A+\cos A)^2+2\cos A+2\sin A-3=0\\\\(\sin A+\cos A)^2+2(\cos A+\sin A)-3=0\\\\\\t^2+2t-3=0\\\\t^2+2t+1-4=0\\\\(t+1)^2=4\\\\t+1=2\qquad\vee \qquad t+1=-2\\\\\\\boxed{t=1\qquad\vee\qquad t=-3}`

When
`A\in\mathbb{R}`

`t=\cos A+\sin A\in[-√(2);√(2)]\approx[-1.4142;1.4142]`

and there is only one answer t = 1.

For
`A\in\mathbb{C}` both values are correct.