Solve the equation 2(sinx+cosx)+sin2x+1=0

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2(sinx+cosx) + sin2x + 1 = 0

We know that sin2x = 2sinxcosx

==> 2(cosx+sinx) + 2sinxcosx + 1 = 0

==> 2(cosx + sinx) + 2sinxcosx + 1= 0

Now we know that sin^2 x + cos^2 x = 1

==> 2(cosx + sinx) + 2sinxcosx + sin^2 x + cos^2 x = 0

==> Now we will rearrange terms.

==> cos^2 x + 2sinxcosx + sin^2 x + 2(sinx+cosx) = 0

==> (cosx+sinx)^2 + 2(sinx+cosx) = 0

Now we will factor (sinx+cosx)

==> (sinx+cosx)(sinx+cosx + 2) = 0

==> sinx + cosx = 0 ==> sinx= -cosx ==> x = 45 in the 2nd, 3rd, and 4th quadrants.

**==> x = 135, 225, and 315.**

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