(a) Find \(\int_0^\pi \sin\theta\,\mathrm{d}\theta\).

(b) Given that \(0\lt \alpha\lt \frac{\pi}{2}\), show that \(\frac{\sec\alpha}{\cot\alpha+\tan\alpha}\) can be written as \(\sin\alpha\).

Show that \(\tan \theta+\cot \theta\) can be written as \(\sec \theta \operatorname{cosec} \theta\).

(a) Show that

\(\frac{\sin x\tan x}{1-\cos x}=1+\operatorname{sec} x.\)

(b) Solve the equation

\(5\tan x-3\operatorname{cot} x=2\operatorname{sec} x\)

for \(0^\circ\leq x\leq360^\circ\).

(i) Show that

\(\frac{\operatorname{cosec}\theta}{\operatorname{cot}\theta+\tan\theta}=\cos\theta.\)

It is given that

\(\int_0^a \frac{\operatorname{cosec}2\theta}{\cot2\theta+\tan2\theta}\,d\theta=\frac{\sqrt3}{4}, \qquad 0\lt a\lt \frac{\pi}{4}.\)

(ii) Using your answer to part (i), find \(a\) in terms of \(\pi\).

(a) Show that

\(\frac{\tan^2\theta+\sin^2\theta}{\cos\theta+\operatorname{sec}\theta}=\tan\theta\sin\theta.\)

(b) Given that \(x=3\sin\phi\) and \(y=\dfrac3{\cos\phi}\), find the numerical value of \(9y^2-x^2y^2\).

(i) Prove that \(\sin x(\operatorname{cot} x+\tan x)=\operatorname{sec} x\).

(ii) Hence solve the equation \(|\sin x(\operatorname{cot} x+\tan x)|=2\) for \(0^\circ\le x\le360^\circ\).