If \(\sin(\theta + \alpha) = a\) and \(\sin(\theta + \beta) = b\), then prove that \(\cos 2(\alpha - \beta) - 4ab \cos(\alpha - \beta) = 1 - 2a^2 - 2b^2\).
If \(\cos(\theta + \phi) = m \cos(\theta - \phi)\), then prove that \(\tan \theta = \dfrac{1 - m}{1 + m} \cot \phi\).
Find the value of the expression
\(3\left[\sin^4\left(\dfrac{3\pi}{2} - \alpha\right) + \sin^4(3\pi + \alpha)\right] - 2\left[\sin^6\left(\dfrac{\pi}{2} + \alpha\right) + \sin^6(5\pi - \alpha)\right].\)
1
If an equation \(a \cos 2\theta + b \sin 2\theta = c\) has \(\alpha\) and \(\beta\) as its roots, then prove that
\(\tan \alpha + \tan \beta = \dfrac{2b}{a + c}.\)
If \(x = \sec \phi - \tan \phi\) and \(y = \csc \phi + \cot \phi\), then show that \(xy + x - y + 1 = 0\).
If \(\theta\) lies in the first quadrant and \(\cos \theta = \dfrac{8}{17}\), then find the value of
\(\cos(30^{\circ} + \theta) + \cos(45^{\circ} - \theta) + \cos(120^{\circ} - \theta).\)
\(\dfrac{23}{17}\left(\dfrac{\sqrt{3} - 1}{2} + \dfrac{1}{\sqrt{2}}\right)\)
Find the value of the expression
\(\cos^4\dfrac{\pi}{8} + \cos^4\dfrac{3\pi}{8} + \cos^4\dfrac{5\pi}{8} + \cos^4\dfrac{7\pi}{8}.\)
\(\dfrac{3}{2}\)
Find the general solution of the equation \(5\cos^2\theta + 7\sin^2\theta - 6 = 0\).
\(n\pi \pm \dfrac{\pi}{4}\)
Find the general solution of the equation
\(\sin x - 3 \sin 2x + \sin 3x = \cos x - 3 \cos 2x + \cos 3x.\)
\(\dfrac{n\pi}{2} \pm \dfrac{\pi}{8}\)
Find the general solution of the equation
\((\sqrt{3} - 1)\cos \theta + (\sqrt{3} + 1)\sin \theta = 2.\)
\(\theta = 2n\pi \pm \dfrac{\pi}{4} + \dfrac{\pi}{12}\)