Prove that \(\dfrac{\tan A + \sec A - 1}{\tan A - \sec A + 1} = \dfrac{1 + \sin A}{\cos A}\).
If \(\dfrac{2\sin\alpha}{1 + \cos\alpha + \sin\alpha} = y\), then prove that \(\dfrac{1 - \cos\alpha + \sin\alpha}{1 + \sin\alpha}\) is also equal to \(y\).
If \(m\sin\theta = n\sin(\theta + 2\alpha)\), then prove that \(\tan(\theta + \alpha)\cot\alpha = \dfrac{m + n}{m - n}\).
If \(\cos(\alpha + \beta) = \dfrac{4}{5}\) and \(\sin(\alpha - \beta) = \dfrac{5}{13}\), where \(\alpha\) lies between \(0\) and \(\dfrac{\pi}{4}\), find the value of \(\tan 2\alpha\).
If \(\tan x = \dfrac{b}{a}\), then find the value of \(\sqrt{\dfrac{a + b}{a - b}} + \sqrt{\dfrac{a - b}{a + b}}\).
Prove that \(\cos\theta\cos\dfrac{\theta}{2} - \cos3\theta\cos\dfrac{9\theta}{2} = \sin7\theta\sin8\theta\).
If \(a\cos\theta + b\sin\theta = m\) and \(a\sin\theta - b\cos\theta = n\), then show that \(a^2 + b^2 = m^2 + n^2\).
Find the value of \(\tan 22^{\circ}30'\).
Prove that \(\sin 4A = 4\sin A\cos^{3}A - 4\cos A\sin^{3}A\).
If \(\tan\theta + \sin\theta = m\) and \(\tan\theta - \sin\theta = n\), then prove that \(m^2 - n^2 = 4\sin\theta\tan\theta\).
If \(\tan(A + B) = p\) and \(\tan(A - B) = q\), then show that \(\tan 2A = \dfrac{p + q}{1 - pq}\).
If \(\cos\alpha + \cos\beta = 0 = \sin\alpha + \sin\beta\), then prove that \(\cos 2\alpha + \cos 2\beta = -2\cos(\alpha + \beta)\).
If \(\dfrac{\sin(x + y)}{\sin(x - y)} = \dfrac{a + b}{a - b}\), then show that \(\dfrac{\tan x}{\tan y} = \dfrac{a}{b}\).
If \(\tan\theta = \dfrac{\sin\alpha - \cos\alpha}{\sin\alpha + \cos\alpha}\), then show that \(\sin\alpha + \cos\alpha = \sqrt{2}\cos\theta\).
If \(\sin\theta + \cos\theta = 1\), then find the general value of \(\theta\).
Find the most general value of \(\theta\) satisfying the equations \(\tan\theta = -1\) and \(\cos\theta = \dfrac{1}{\sqrt{2}}\).
If \(\cot\theta + \tan\theta = 2\csc\theta\), then find the general value of \(\theta\).
If \(2\sin^{2}\theta = 3\cos\theta\), where \(0 \le \theta \le 2\pi\), then find the value(s) of \(\theta\).
If \(\sec x \cos 5x + 1 = 0\), where \(0 < x \le \dfrac{\pi}{2}\), then find the value of \(x\).