Given y = $\mathit{t}\mathit{a}{\mathit{n}}^{-1}\left(\frac{1+\mathit{c}\mathit{o}\mathit{s}\mathit{x}}{\mathit{s}\mathit{i}\mathit{n}\mathit{x}}\right)$
⇒ y = $\mathit{t}\mathit{a}{\mathit{n}}^{-1}$ $\left(\frac{2\mathit{c}\mathit{o}{\mathit{s}}^2\frac{\mathit{x}}{2}}{2\mathit{s}\mathit{i}\mathit{n}\frac{\mathit{x}}{2}.\mathit{c}\mathit{o}\mathit{s}\frac{\mathit{x}}{2}}\right)$ (Since 1 + cos x = 2 $\mathit{c}\mathit{o}{\mathit{s}}^2\frac{\mathit{x}}{2}$ and sin x = 2 sin $\frac{\mathit{x}}{2}$ cos $\frac{\mathit{x}}{2}$)
⇒ y = $\mathit{t}\mathit{a}{\mathit{n}}^{-1}$ $\left(\cot \frac{\mathit{x}}{2}\right)$
⇒ y = $\mathit{t}\mathit{a}{\mathit{n}}^{-1}$ $\left[\mathit{t}\mathit{a}\mathit{n}\left(\frac{\mathit{\pi }}{2}-\frac{\mathit{x}}{2}\right)\right]$
y = $\frac{\mathit{\pi }}{2}-\frac{\mathit{x}}{2}$ [Since $\mathit{t}\mathit{a}{\mathit{n}}^{-1}$ (tan x) = x]
Differentiating with respect to x,
⇒ $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ = 0 - $\frac{1}{2}$ = - $\frac{1}{2}$