Is 1 - sin(x) = cos(x)

Nope, because if x is for example superior to Pi/2, let's say it equals Pi, then:
cos(x) = -1
sin(x)=0

So no, it doesn't work.

No.

Edit: 1 - sin^2(x) = cos^2(x)

1 - sin(x) = cos(x) => for x = 0 , pi/2

Edit: 1 - sin^2(x) = cos^2(x)

1 - sin(x) = cos(x) => for x = 0 , pi/2

Hell no! 1 – sin²x = cos²x. We would have no use for both functions if the relationship were so simple

No it's not