Simplifying $\frac{\omega\sin(\omega t)\tan(\omega{t})+\omega(\cos(\omega t)+1)\sec^2(\omega t)}{(\cos(\omega t) +1)^2}$ How can I simply this :
$$
\dfrac{{\omega}\sin \left( {\omega} {t}\right) \tan \left( {\omega}{t}\right) +\omega \left( \cos \left( \omega t\right) +1\right) \sec ^{2}\left( \omega t\right) }{\left( \cos \left( \omega t\right) +1\right) ^{2}}
$$
To become this :
$$
\dfrac{\omega \left( -\cos ^{2}\left( \omega t\right) +\cos \left( \omega t\right) +1\right) }{1+\cos \left( \omega t\right) }
$$
 A: $$
\dfrac{\omega\sin \left( \omega t\right) \tan \left( \omega{t}\right) +\omega \left( \cos \left( \omega t\right) +1\right) \sec ^{2}\left( \omega t\right) }{\left( \cos \left( \omega t\right) +1\right) ^{2}},
\\
\sec(\omega{t})=\frac{1}{\cos(\omega{t})},\quad\tan(\omega{t})=\frac{\sin(\omega{t})}{\cos(\omega{t})},\implies
\\
\implies\dfrac{\omega\sin \left( \omega t\right) \tan \left( \omega{t}\right) +\omega \left( \cos \left( \omega t\right) +1\right) \sec ^{2}\left( \omega t\right) }{\left( \cos \left( \omega t\right) +1\right) ^{2}}=\frac{\omega\frac{\sin^{2}(\omega{t})}{\cos(\omega{t})}+\omega\left(\frac{\cos(\omega{t})+1}{\cos^{2}(\omega{t})}\right)}{\left(\cos\left(\omega{t}\right)+1\right)^{2}}=
\\
=\frac{\omega\frac{\sin^{2}(\omega{t})\cos(\omega{t})}{\cos^{2}(\omega{t})}+\omega\left(\frac{\cos(\omega{t})+1}{\cos^{2}(\omega{t})}\right)}{\left(\cos\left(\omega{t}\right)+1\right)^{2}}=
\frac{\omega(2\cos(\omega{t})-\cos^{3}(\omega{t})+1)}{\bbox[lightgreen]{\cos^{2}(\omega{t})}(\cos(\omega{t})+1)^{2}}.\qquad\qquad(1)
$$
If we will multiply that we must get on the expression $\frac{(1+\cos(\omega{t})}{(1+\cos(\omega{t})}$, therefore, the result will be:
$$
\dfrac{\omega \left( -\cos ^{2}\left( \omega t\right) +\cos \left( \omega t\right) +1\right) }{1+\cos \left( \omega t\right) }=\frac{\omega\left( -\cos ^{2}\left( \omega t\right) +\cos \left( \omega t\right) +1-\cos^{3}(\omega{t})+\cos^{2}(\omega{t})+\cos(\omega{t})\right)}{(1+\cos(\omega{t}))^{2}}=
\\
=\frac{\omega\left(1-\cos^{3}(\omega{t})+2\cos(\omega{t})\right)}{(1+\cos(\omega{t}))^{2}}.\qquad\qquad(2)
$$
As we can see in $(2)$ the multiplier $\bbox[lightgreen]{\cos^{2}(\omega{t})}$ is absent, therefore, the misprint takes a place.
