(i)[3]
By differentiating $\frac{1}{\sin x}$, demonstrate that if $y = \cosec x$ then $\frac{dy}{dx} = -\cosec x \cot x$.
(ii)[2]
By equating the gradients of the curves at $x = a$, establish that $a = \tan^{-1}\!\left(\frac{e^a}{\sin a}\right)$.
(iii)[2]
Use calculation to confirm that $a$ is between 1 and 1.5.
(iv)[3]
Apply the iterative formula derived from the equation in part (ii) to find $a$ correct to 3 decimal places. Record each iterate to 5 decimal places.