Solve the equation sin x + cos x + sin x cos x = 1

Question

I know the answer is sin x = 1 or cos x = 1, but how do I do the working?

Answer 1

sinx + cosx + sinx cosx = 1

=> sinxx + cosx = 1 - sinx cosx

squaring both sides

1 + 2sinx cosx = 1 - 2sinx cosx + sinx*sinx*cosx*cosx

=> sinx*sinx*cosx*cosx - 4sinx cosx = 0

=> sinx cosx (sinx cosx - 4) = 0

=> sinx = 0 or cosx = 0 as sinxcosx < 1/2

Plugging sinx=0 in the original equation yields: cosx = 1

Similarly, Plugging cosx=0 in the original equation yields: sinx= 1