If $$y\left( \alpha \right) = \sqrt {2\left( {{{\tan \alpha + \cot \alpha } \over {1 + {{\tan }^2}\alpha }}} \right) + {1 \over {{{\sin }^2}\alpha }}} ,\alpha \in \left( {{{3\pi } \over 4},\pi } \right)$$

${{dy} \over {d\alpha }}\,\,at\,\alpha = {{5\pi } \over 6}is$ :

$$y\left( \alpha \right) = \sqrt {2\left( {{{\tan \alpha + \cot \alpha } \over {1 + {{\tan }^2}\alpha }}} \right) + {1 \over {{{\sin }^2}\alpha }}}$$ <br><br>= $$\sqrt {2\left( {{{1 + {{\tan }^2}\alpha } \over {\tan \alpha \left( {1 + {{\tan }^2}\alpha } \right)}}} \right) + {1 \over {{{\sin }^2}\alpha }}} $$ <br><br>= $$\sqrt {2\left( {{{\cos \alpha } \over {\sin \alpha }}} \right) + {1 \over {{{\sin }^2}\alpha }}} $$ <br><br>= $\sqrt {{{\sin 2\alpha + 1} \over {{{\sin }^2}\alpha }}}$ <br><br>= $${{\left| {\sin \alpha + \cos \alpha } \right|} \over {\left| {\sin \alpha } \right|}}$$ <br><br>At $\alpha$ = ${{5\pi } \over 6}$ <br><br>${\left| {\sin \alpha + \cos \alpha } \right|}$ = -(sin$\alpha$ + cos$\alpha$) <br><br>and |sin$\alpha$| = sin$\alpha$ <br><br>$\therefore$ y($\alpha$) = ${{ - \left( {\sin \alpha + \cos \alpha } \right)} \over {\sin \alpha }}$ <br><br>= -1 - cot$\alpha$ <br><br>$\therefore$ ${{dy} \over {d\alpha }}$ = cosec²$\alpha$ <br><br>So ${{{dy} \over {d\alpha }}}$ at $\alpha$ = ${{5\pi } \over 6}$, <br><br>= cosec²${{5\pi } \over 6}$ = 4