Using the Chain Rule

Differentiating \(y = \cos^2\theta\) using the chain rule gives:
\begin{eqnarray*}  
y = z^2 && z = \cos\theta\\ 
\frac{\mathrm{d}y}{\mathrm{d}z} = 2z &&\frac{\mathrm{d}z}{\mathrm{d}\theta} = -\sin \theta\\\\ 
\frac{\mathrm{d}y}{\mathrm{d}\theta} &=& \frac{\mathrm{d} y}{\mathrm{d} z} \times \frac{\mathrm{d} z}{\mathrm{d}\theta}\\ 
\frac{\mathrm{d} y}{\mathrm{d}\theta} &=& 2z\times (-\sin\theta) \\  
&=& -2\cos\theta\sin\theta\\  
&=& -\sin 2\theta 
\end{eqnarray*}