$\frac{\mathit{\ⅆ}}{\mathit{\ⅆ}u} {\left(\frac{\cos \left(u\right)}{\sin \left(2 u\right)}\right)}^2$ = $-\frac{2\cos \left(u\right)\sin \left(u\right)}{{\sin \left(2u\right)}^2}-\frac{4{\cos \left(u\right)}^2\cos \left(2u\right)}{{\sin \left(2u\right)}^3}$$\stackrel{\text{simplify}}{\=}$$-\frac{1}{2}\frac{\cos \left(u\right)}{{\sin \left(u\right)}^3}$$$

$\frac{d}{\mathrm{du}}$ ${\(\frac{\cos \(u\)}{\sin \(2u\)}\)}^2$

$\=\frac{2\cos \(u\)\(\frac{\mathit{\ⅆ}}{\mathit{\ⅆ}u}\(\frac{\cos \(u\)}{\sin \(2u\)}\)\)}{\sin \(2u\)}$

$\=2\frac{\cos \(u\)}{\sin \(2u\)}$  $\frac{\sin \(2u\)\(\frac{\mathit{\ⅆ}}{\mathit{\ⅆ}u}\cos \(u\)\)-\cos \(u\)\(\frac{\mathit{\ⅆ}}{\mathit{\ⅆ}u}\sin \(2u\)\)}{{\sin \(2u\)}^2}$

$\=2\frac{\cos \(u\)}{\sin \(2u\)}\(\frac{\sin \(2u\)\left(-\sin \(u\)\right)-\cos \(u\)2\cos \(2u\)}{{\sin \(2u\)}^2}\)$  

$\=-\frac{2\cos \(u\)\(\sin \(2u\)\sin \(u\)\+2\cos \(u\)\cos \(2u\)\)}{{\sin \(2u\)}^3}$

Table 2.7.2(a)   First solution for the evaluation of $\frac{d}{\mathrm{du}} {\left(\frac{\cos \left(u\right)}{\sin \left(2 u\right)}\right)}^2$ 

$\frac{d}{\mathrm{du}}$  ${\(\frac{\cos \(u\)}{\sin \(2u\)}\)}^2$

$\=-\frac{2\cos \(u\)\(2\sin \(u\)\cos \(u\)\sin \(u\)\+2\cos \(u\)\({\cos }^{2}u-{\sin }^{2}u\)\)}{{\sin \(2u\)}^3}$

$\=-\frac{4{\cos }^{2}u\({\sin }^{2}u\+{\cos }^{2}u-{\sin }^{2}u\)}{{\sin }^3\(2u\)}$

$\=-\frac{4{\cos }^{2}u{\cos }^{2}u}{{\sin }^3\(2u\)}$

$\=-\frac{4{\cos }^{4}u}{{\sin }^3\(2u\)}$

$\= -\frac{4 {\cos }^4\left(u\right)}{{\left(2 \sin \left(u\right)\cos \left(u\right)\right)}^2}$

$\= -\frac{4 {\cos }^4\left(u\right)}{8 {\sin }^3\left(u\right){\cos }^3\left(u\right)}$

$\= -\frac{\cos \left(u\right)}{2 {\sin }^3\left(u\right)}$

$\= -\frac{1}{2} \frac{1}{{\sin }^2\left(u\right)} \frac{\cos \left(u\right)}{\sin \left(u\right)}$

$\= -\frac{1}{2} \frac{\cos \left(u\right)}{{\sin }^3\left(u\right)}$

Table 2.7.2(b)   Further simplifications of the derivative in Table 2.6.13

$\frac{d}{\mathrm{du}} {\left(\frac{\cos \left(u\right)}{\sin \left(2u\right)}\right)}^2$

$\=\frac{d}{\mathrm{du}} {\left(\frac{\cos \left(u\right)}{2\sin \left(u\right)\cos \left(u\right)}\right)}^2$

$\= \frac{d}{\mathrm{du}} \frac{1}{{\left(2\sin \left(u\right)\right)}^2}$

$\= \frac{d}{\mathrm{du}}$$\frac{{\sin \left(u\right)\)}^{-2}}{4}$

$\= \frac{d}{\mathrm{du}} \frac{{\csc }^2\left(u\right)}{4}$

$\=\frac{2}{4} \csc \left(u\right)\left(-\csc \left(u\right) \cot \left(u\right)\right)$

$\= -\frac{1}{2} {\csc }^2\left(u\right) \cot \left(u\right)$

$\= -\frac{1}{2} \frac{\cos \left(u\right)}{{\sin }^3\left(u\right)}$

Table 2.7.2(c)   Differentiation after first applying a trig simplification