Difference Between Right and Wrong

WRONG: $\frac{1}{(x^{2}+1)\sin x}\cdot\left((x^{2}+1)\cos x+(\sin x)\cdot2x\right)=\cos x+2x=2x\cos x$

You have to factor or distribute before you can cancel terms.

CORRECT: $\frac{1}{(x^{2}+1)\sin x}\cdot\left((x^{2}+1)\cos x+(\sin x)\cdot2x\right)=$
$\frac{(x^{2}+1)\cos x}{(x^{2}+1)\sin x}+\frac{2x\cdot\sin x}{(x^{2}+1)\sin x}=\frac{\cos x}{\sin x}+\frac{2x}{x^{2}+1}=\cot x+\frac{2x}{x^{2}+1}$

WRONG: $\frac{1}{2xyy^{\prime}+y^{2}}=\frac{1}{2xyy^{\prime}}+\frac{1}{y^{2}}$

CORRECT: Well... avoid that mistake.

WRONG: $2\sin4x^{2}=8\sin x^{2}$

CORRECT: Well... avoid that mistake.

WRONG: $\frac{\sin(2x^{2})(4x)}{x}=\sin(2x^{2})(3x)$

CORRECT: $\frac{\sin(2x^{2})(4x)}{x}=\sin(2x^{2})(4)=4\sin(2x^{2})$

WRONG: $\cos0=0$

CORRECT: $\cos0=1$

WRONG: $\cos=1$

CORRECT: Er... cosine is a function, so it should be $\cos x$ or
$\cos u$ etc...