Express the equation \(\frac{5 + 2 \tan x}{3 + 2 \tan x} = 1 + \tan x\) as a quadratic equation in \(\tan x\) and hence solve the equation for \(0 \leq x \leq \pi\).
(i) Show that the equation \(\frac{\cos \theta + 4}{\sin \theta + 1} + 5 \sin \theta - 5 = 0\) may be expressed as \(5 \cos^2 \theta - \cos \theta - 4 = 0\).
(ii) Hence solve the equation \(\frac{\cos \theta + 4}{\sin \theta + 1} + 5 \sin \theta - 5 = 0\) for \(0^\circ \leq \theta \leq 360^\circ\).
(i) Show that the equation \(\cos 2x(\tan^2 2x + 3) + 3 = 0\) can be expressed as \(2 \cos^2 2x + 3 \cos 2x + 1 = 0\).
(ii) Hence solve the equation \(\cos 2x(\tan^2 2x + 3) + 3 = 0\) for \(0^\circ \leq x \leq 180^\circ\).
(i) Show that the equation \((\sin \theta + 2 \cos \theta)(1 + \sin \theta - \cos \theta) = \sin \theta(1 + \cos \theta)\) may be expressed as \(3 \cos^2 \theta - 2 \cos \theta - 1 = 0\).
(ii) Hence solve the equation \((\sin \theta + 2 \cos \theta)(1 + \sin \theta - \cos \theta) = \sin \theta(1 + \cos \theta)\) for \(-180^\circ \leq \theta \leq 180^\circ\).
(i) Show that \(\cos^4 x \equiv 1 - 2\sin^2 x + \sin^4 x\).
(ii) Hence, or otherwise, solve the equation \(8\sin^4 x + \cos^4 x = 2\cos^2 x\) for \(0^\circ \leq x \leq 360^\circ\).
(i) Show that \(3 \sin x \tan x - \cos x + 1 = 0\) can be written as a quadratic equation in \(\cos x\) and hence solve the equation \(3 \sin x \tan x - \cos x + 1 = 0\) for \(0 \leq x \leq \pi\).
(ii) Find the solutions to the equation \(3 \sin 2x \tan 2x - \cos 2x + 1 = 0\) for \(0 \leq x \leq \pi\).
(a) Show that the equation
\(4 \sin x + \frac{5}{\tan x} + \frac{2}{\sin x} = 0\)
may be expressed in the form \(a \cos^2 x + b \cos x + c = 0\), where \(a, b\) and \(c\) are integers to be found.
(b) Hence solve the equation \(4 \sin x + \frac{5}{\tan x} + \frac{2}{\sin x} = 0\) for \(0^\circ \leq x \leq 360^\circ\).
Solve the equation \(3 \sin^2 \theta = 4 \cos \theta - 1\) for \(0^\circ \leq \theta \leq 360^\circ\).
Show that the equation \(\frac{1}{\cos \theta} + 3 \sin \theta \tan \theta + 4 = 0\) can be expressed as \(3 \cos^2 \theta - 4 \cos \theta - 4 = 0\), and hence solve the equation \(\frac{1}{\cos \theta} + 3 \sin \theta \tan \theta + 4 = 0\) for \(0^\circ \leq \theta \leq 360^\circ\).
(i) Show that the equation \(\frac{4 \cos \theta}{\tan \theta} + 15 = 0\) can be expressed as \(4 \sin^2 \theta - 15 \sin \theta - 4 = 0\).
(ii) Hence solve the equation \(\frac{4 \cos \theta}{\tan \theta} + 15 = 0\) for \(0^\circ \leq \theta \leq 360^\circ\).
(i) Prove the identity \(\frac{\sin \theta - \cos \theta}{\sin \theta + \cos \theta} \equiv \frac{\tan \theta - 1}{\tan \theta + 1}\).
(ii) Hence solve the equation \(\frac{\sin \theta - \cos \theta}{\sin \theta + \cos \theta} = \frac{\tan \theta}{6}\), for \(0^\circ \leq \theta \leq 180^\circ\).
(i) Show that the equation \(1 + \sin x \tan x = 5 \cos x\) can be expressed as \(6 \cos^2 x - \cos x - 1 = 0\).
(ii) Hence solve the equation \(1 + \sin x \tan x = 5 \cos x\) for \(0^\circ \leq x \leq 180^\circ\).
(i) Solve the equation \(4 \sin^2 x + 8 \cos x - 7 = 0\) for \(0^\circ \leq x \leq 360^\circ\).
(ii) Hence find the solution of the equation \(4 \sin^2 \left(\frac{1}{2} \theta\right) + 8 \cos \left(\frac{1}{2} \theta\right) - 7 = 0\) for \(0^\circ \leq \theta \leq 360^\circ\).
(i) Express the equation \(2 \cos^2 \theta = \tan^2 \theta\) as a quadratic equation in \(\cos^2 \theta\).
(ii) Solve the equation \(2 \cos^2 \theta = \tan^2 \theta\) for \(0 \leq \theta \leq \pi\), giving solutions in terms of \(\pi\).
Solve the equation \(7 \cos x + 5 = 2 \sin^2 x\), for \(0^\circ \leq x \leq 360^\circ\).
(i) Show that the equation \(2 \cos x = 3 \tan x\) can be written as a quadratic equation in \(\sin x\).
(ii) Solve the equation \(2 \cos 2y = 3 \tan 2y\), for \(0^\circ \leq y \leq 180^\circ\).
(i) Solve the equation \(2 \cos^2 \theta = 3 \sin \theta\), for \(0^\circ \leq \theta \leq 360^\circ\).
(ii) The smallest positive solution of the equation \(2 \cos^2(n\theta) = 3 \sin(n\theta)\), where \(n\) is a positive integer, is \(10^\circ\). State the value of \(n\) and hence find the largest solution of this equation in the interval \(0^\circ \leq \theta \leq 360^\circ\).
(a) Show that the equation
\(3 \tan^2 x - 3 \sin^2 x - 4 = 0\)
may be expressed in the form \(a \cos^4 x + b \cos^2 x + c = 0\), where \(a, b\) and \(c\) are constants to be found.
(b) Hence solve the equation \(3 \tan^2 x - 3 \sin^2 x - 4 = 0\) for \(0^\circ \leq x \leq 180^\circ\).
(i) Prove the identity \(\tan x + \frac{1}{\tan x} \equiv \frac{1}{\sin x \cos x}\).
(ii) Solve the equation \(\frac{2}{\sin x \cos x} = 1 + 3 \tan x\), for \(0^\circ \leq x \leq 180^\circ\).
(i) Given that
\(3 \sin^2 x - 8 \cos x - 7 = 0\),
show that, for real values of \(x\),
\(\cos x = -\frac{2}{3}\).
(ii) Hence solve the equation
\(3 \sin^2(\theta + 70^\circ) - 8 \cos(\theta + 70^\circ) - 7 = 0\)
for \(0^\circ \leq \theta \leq 180^\circ\).