Mathematics - Differentiation Question with Solution | TestHub

$\begin{array}{rl}& y=\frac{2\cos \theta +2{\cos }^2\theta -1}{4{\cos }^3\theta -3\cos \theta +8{\cos }^2\theta -4+5\cos \theta +2}\\ & y=\frac{\left(2{\cos }^2\theta +2\cos \theta -1\right)}{\left(2{\cos }^2\theta +2\cos \theta -1\right)(2\cos \theta +2)}\\ & y=\frac{1}{2}\left(\frac{1}{1+\cos \theta }\right)\\ & \Rightarrow \theta =\frac{\pi }{2}y=\frac{1}{2}\\ & y^{\prime }=\frac{1}{2}\left(\frac{-1}{(1+\cos \theta {)}^2}\times (-\sin \theta )\right)\\ & \Rightarrow \theta =\frac{\pi }{2}y=\frac{1}{2}\end{array}$ $\begin{array}{rl}{y}^{\prime \prime }& =\frac{1}{2}\left[\frac{\cos \theta (1+\cos \theta {)}^2-\sin \theta (2)(1+\cos \theta )(-\sin \theta )}{(1+\cos \theta {)}^4}\right]\\ & \Rightarrow \theta =\frac{\pi }{2}y=1\end{array}$