Simple Laplacian operator on the sphere $\mathbb{S}^2$

Question

Let

\begin{cases} x=\sin \theta \cos \varphi \\ y= \sin \theta \sin \varphi \\ z= \cos \theta \end{cases}

Then we obtain

$$g_{\mathbb{S}^2} = \begin{pmatrix} 1& 0 \\ 0 & \sin^2 \theta \end{pmatrix}$$

Hence, $$\det (g_{\mathbb{S}^2})= \sin^2 \theta$$

Furthermore,

$$\Delta_{\mathbb{S}^2} = \frac{1}{\sqrt{\det (g_{\mathbb{S}^2})}} \sum_{i,j} \frac{\partial}{\partial \theta_i} (\sqrt{\det (g_{\mathbb{S}^2})} g_{ij} \frac{\partial}{\partial \theta_j}) = \frac{1}{ \sin \theta} (\frac{\partial}{\partial \theta} (\sin \theta \frac{\partial}{\partial \theta}) + \frac{\partial}{\partial \varphi} (\sin \theta \frac{\partial}{\partial \varphi})) $$

$$= \frac{1}{ \sin \theta} \frac{\partial}{\partial \theta} (\sin \theta \frac{\partial}{\partial \theta})+ \frac{\partial^2}{\partial \varphi^2}. $$

As final answer, I'm supposed to obtain $\frac{1}{ \sin \theta} \frac{\partial}{\partial \theta} (\sin \theta \frac{\partial}{\partial \theta})+ \frac{1}{\sin^2 \theta} \frac{\partial^2}{\partial \varphi^2}$, but I just don't know where is my mistake.

Thanks!