1 A curve $C$ has equation $\cos y = x$, for $- \pi < x < \pi$.\\
(i) Use implicit differentiation to show that

$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = - \cot y \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 }$$

(ii) Hence find the exact value of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ at the point $\left( \frac { 1 } { 2 } , \frac { 1 } { 3 } \pi \right)$ on $C$.\\

\hfill \mbox{\textit{CAIE FP1 2019 Q1}}