Proofs That Go Wrong

Four moves look like algebra and are not:

Cancelling a term of a sum. In $\frac{\sin\left(x\right)+\cos\left(x\right)}{\sin\left(x\right)}$ the sin x cannot be crossed out: it is a term on top, not a factor. Split the fraction instead: $1+\frac{\cos\left(x\right)}{\sin\left(x\right)}$.

Splitting a function. $\sin\left(2x\right)$ is not $2\sin\left(x\right)$, and $\sin\left(x+y\right)$ is not $\sin\left(x\right)+\sin\left(y\right)$: a function is not a number to multiply out.

Reading the square as the angle. ${\sin\left(x\right)}^{2}$ squares the value, $\sin\left({x}^{2}\right)$ squares the angle.

Dividing by something that can be 0. Dividing by sin x is not allowed where sin x is 0; factor instead.