Решить уравнение sin x + cos x = 1

2sin (x/2) cos (x/2) + cos² (x/2) - sin² (x/2) - sin² (x/2) - cos² (x/2) = 0

2sin (x/2) cos (x/2) - 2sin² (x/2) = 0

2sin (x/2) * (cos (x/2) - sin (x/2)) = 0

sin (x/2) = 0⇒x/2=πn⇒x=2πn, n∈z

cos (x/2) - sin (x/2) = 0/cos (x/2)

1-tg (x/2) = 0

tg (x/2) = 1⇒x/2=π/4+πk⇒x=π/2+2πk, k∈z

Возведем обе части в квадрат

(sinx+cosx) ²=1

sin²x+2sinxcosx+cos²x=1

1+sin2x=1

sin2x=0

2x=πk, k∈Z

x=πk/2, k∈Z