What's wrong with my derivative?

[plucky kraken]

If I try to differentiate (sin(x))^2, I can present it in form of (sin(x)sin(x))' and if so I can use product rule resulting in:
sin(x)(sin(x))'+(sin(x))'*sin(x)
=sin(x)cos(x)+sin(x)cos(x)
=2sin(x)cos(x)
=sin(2x)
But integral of sin(2x) of course is not equal to (sin(x))^2 so where is the problem?

[forest viper]

there is no problem

german?

[plucky kraken]

So sin(2x)/2 is equal to (sin(x))^2? If it is, my bad)

[plucky kraken]

Nope

[plucky kraken]

Oh, it's cosine

[forest viper]

The integral of sin(2x) is -cos(2x)/2 + C

[plucky kraken]

Really my bad

Thanks:)

[forest viper]

And if you compare -cos(2x)/2 + C to sin²(x) the integral only differs by some constant C