(a) The diagram shows an equilateral triangle \(ABC\) with side \(a\). \(M\) is the midpoint of \(AC\) and angle \(AMB=90^\circ\).

Use the diagram to find \(\sec30^\circ\).

(b) Show that \(\dfrac{1}{\sec x-1}+\dfrac{1}{\sec x+1}\) can be written as \(2\operatorname{cosec}x\cot x\).

For \(-1\leqslant x\leqslant1\), it is given that \(x=2-3\sin\theta\) and \(y=3\cot^2\theta\).

Find \(y\) in terms of \(x\).

(a) Given that \(0\leqslant\theta\lt \frac{\pi}{2}\), show that \(\frac{\sin\theta}{\sqrt{\operatorname{cosec}^2\theta-1}}+\frac{1}{\sqrt{1+\tan^2\theta}}\) can be written as \(\sec\theta\).

(b) Given that \(\sec x=\alpha\), where \(\frac{3\pi}{2}\lt x\leqslant2\pi\), find \(\sin x\) in terms of \(\alpha\).

(a) Show that \((2 \tan \theta+\sec \theta)(2 \tan \theta-\sec \theta)=3 \tan ^{2} \theta-1\).

(b) Hence solve the equation \((2 \tan \theta+\sec \theta)(2 \tan \theta-\sec \theta)=1\) for \(0^{\circ} \leqslant \theta \leqslant 180^{\circ}\).

Given that \(2+\cot \theta=3 x\) and \(\sin \theta=\sqrt{y}\), find \(y\) in terms of \(x\).

(a) Show that \((\tan x+\sec x)^{2}\) can be written as \(\frac{1+\sin x}{1-\sin x}\). (b) Hence solve the equation \((\tan 3 \theta+\sec 3 \theta)^{2}=6\) for \(0^{\circ} \leqslant \theta \leqslant 180^{\circ}\).

(a) Show that

\(\displaystyle \frac{\sin x}{\tan x-1}-\frac{\cos x}{\tan x+1} =\frac{\cos x}{\sin^2x-\cos^2x}.\)

(b) Hence solve the equation

\(\displaystyle \frac{\sin x}{\tan x-1}-\frac{\cos x}{\tan x+1}=1\)

for \(0^\circ\lt x\lt360^\circ\).

(a) Show that

\(\displaystyle \frac1{\sec x-\cosec x}+\frac1{\sec x+\cosec x}=\frac{2\cos x}{1-\cot^2x}.\)

(b) Solve the equation

\(3\tan^2\left(y+\frac\pi4\right)=1\)

for \(-2\pi\lt y\lt0\).

(a) By writing \(\cot x\) and \(\tan x\) in terms of \(\cos x\) and \(\sin x\), show that

\(\displaystyle \frac{\sin x}{1-\cot x}+\frac{\cos x}{1-\tan x}=\sin x+\cos x.\)

(b) Solve the equation \(9\cot x+3\operatorname{cosec}x=\tan x\), for \(0^\circ\lt x\lt360^\circ\).

(a) Show that

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

(b) Hence solve the equation

\(\displaystyle \left(\frac{\operatorname{cot}\left(\frac{\phi}{3}\right)+\tan\left(\frac{\phi}{3}\right)}{\operatorname{sec}\left(\frac{\phi}{3}\right)}\right)^2=2\)

for \(-540^\circ\lt\phi\lt540^\circ\).

(a) Show that

\(\cos^4\theta-\sin^4\theta+1=2\cos^2\theta.\)

(b) Solve the equation

\(\cos^4\frac{\phi}{3}-\sin^4\frac{\phi}{3}+1=\frac12,\)

for \(-3\pi\lt\phi\lt3\pi\), giving your answers in terms of \(\pi\).

(a) Write down the values of \(k\) for which \(y=k\) is a tangent to \(y=4\sin\left(x+\frac\pi4\right)+10\).

(b)(i) Show that

\(\frac{1+\tan\theta}{1-\cos\theta}+\frac{1-\tan\theta}{1+\cos\theta}=\frac{2(1+\sin\theta)}{\sin^2\theta}.\)

(b)(ii) Hence solve the equation where the left side is equal to \(3\), for \(0^\circ\le\theta\le360^\circ\).

(a) Show that

\(\frac{1}{\operatorname{cosec}\theta-1}+ \frac{1}{\operatorname{cosec}\theta+1} =2\sin\theta\,\operatorname{sec}^2\theta.\)

(b) Hence solve the equation

\(\frac{1}{\operatorname{cosec}2\phi-1}+ \frac{1}{\operatorname{cosec}2\phi+1} =4\sin2\phi,\)

for \(-90^\circ\leq\phi\leq90^\circ\).

(a) Show that

\(\frac{\sin x}{1-\cos x}+\frac{1-\cos x}{\sin x}=2\operatorname{cosec}x.\)

(b) Hence solve the equation

\(\frac{\sin x}{1-\cos x}+\frac{1-\cos x}{\sin x}=3\sin x-1\)

for \(0^\circ\lt x\lt 360^\circ\).

(a) Show that

\(\frac{\cos x}{1-\sin x}+\frac{1-\sin x}{\cos x}\equiv 2\operatorname{sec} x.\)

(b) Hence solve the equation

\(\frac{\cos(\frac12\theta)}{1-\sin(\frac12\theta)}+\frac{1-\sin(\frac12\theta)}{\cos(\frac12\theta)}=8\cos^2(\tfrac12\theta)\)

for \(-360^\circ\lt \theta\lt 360^\circ\).

(a) Solve the equation

\(\sin\alpha\,\operatorname{cosec}^2\alpha+\cos\alpha\,\operatorname{sec}^2\alpha=0\)

for \(-\pi\lt \alpha\lt \pi\), giving your answers in terms of \(\pi\).

(b)

(i) Prove the identity

\(\frac{\cos\theta}{1-\sin\theta}+\frac{1-\sin\theta}{\cos\theta}\equiv 2\operatorname{sec}\theta.\)

(ii) Hence solve the equation

\(\frac{\cos3\phi}{1-\sin3\phi}+\frac{1-\sin3\phi}{\cos3\phi}=4\)

for \(0^\circ\leq\phi\leq180^\circ\).

(a) Show that

\(\frac1{\operatorname{sec}x-1}+\frac1{\operatorname{sec}x+1} =2\operatorname{cot} x\operatorname{cosec}x.\)

(b) Hence solve the equation

\(\frac1{\operatorname{sec}x-1}+\frac1{\operatorname{sec}x+1} =3\operatorname{sec}x\)

for \(0^\circ\lt x\lt 360^\circ\).

(a) Show that

\(\frac1{\operatorname{cosec}x-1}+\frac1{\operatorname{cosec}x+1} =2\tan x\operatorname{sec}x.\)

(b) Hence solve the equation

\(\frac1{\operatorname{cosec}x-1}+\frac1{\operatorname{cosec}x+1} =5\operatorname{cosec}x\)

for \(0^\circ\lt x\lt 360^\circ\).

(a)(i) Show that

\(\frac{1}{\operatorname{sec}\theta-1}-\frac{1}{\operatorname{sec}\theta+1}=2\operatorname{cot}^2\theta.\)

(a)(ii) Hence solve

\(\frac{1}{\sec2x-1}-\frac{1}{\sec2x+1}=6\)

for \(-90^\circ\lt x\lt90^\circ\).

(b) Solve

\(\operatorname{cosec}\left(y+\frac{\pi}{3}\right)=2\)

for \(0\leq y\leq2\pi\) radians, giving your answers in terms of \(\pi\).

(a)(i) Show that

\(\frac{1}{(1+\operatorname{cosec}\theta)(\sin\theta-\sin^2\theta)} =\operatorname{sec}^2\theta.\)

(a)(ii) Hence solve

\((1+\operatorname{cosec}\theta)(\sin\theta-\sin^2\theta)=\frac34\)

for \(-180^\circ\leq\theta\leq180^\circ\).

(b) Solve

\(\sin\left(3\phi+\frac{2\pi}{3}\right)=\cos\left(3\phi+\frac{2\pi}{3}\right)\)

for \(0\leq\phi\leq\dfrac{2\pi}{3}\) radians, giving your answers in terms of \(\pi\).

(a) Solve the equation \(\sin x\cos x=\dfrac12\tan x\), for \(0^\circ\leq x\leq180^\circ\).

(b) (i) Prove the identity

\(\operatorname{sec}\theta-\frac{\sin\theta}{\operatorname{cot}\theta}\equiv\cos\theta.\)

(ii) Hence solve

\(\operatorname{sec}3\theta-\frac{\sin3\theta}{\cot3\theta}=\frac12\)

for \(-\dfrac{2\pi}{3}\leq\theta\leq\dfrac{2\pi}{3}\).

(i) Show that \(\cos\theta\operatorname{cot}\theta+\sin\theta=\operatorname{cosec}\theta\).

(ii) Hence solve \(\cos\theta\operatorname{cot}\theta+\sin\theta=4\) for \(0^\circ\leq\theta\leq90^\circ\).

(i) Show that

\(\frac{1}{1-\cos x}-\frac{1}{1+\cos x}=2\operatorname{cosec}x\operatorname{cot} x.\)

(ii) Hence solve the equation

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

for \(0\leq x\leq2\pi\) radians.

(i) Show that

\(\frac{1}{1-\sin x}-\frac{1}{1+\sin x}=2\tan x\operatorname{sec} x.\)

(ii) Hence solve the equation

\(\frac{1}{1-\sin x}-\frac{1}{1+\sin x}=\operatorname{cosec}x\)

for \(0^\circ\lt x\lt 360^\circ\).

Show that \(\dfrac1{1-\sin\theta}-\dfrac1{1+\sin\theta}=2\tan\theta\operatorname{sec}\theta\).