If $\tan \theta + \sin \theta = m$ and $\tan \theta - \sin \theta = n,$ then

Consider the following parametric equation of a curve: $x(\theta) = |\cos 4\theta| \cos \theta$ and $y(\theta) = |\cos 4\theta| \sin \theta$,where $0 \leq \theta \leq 2\pi$. Which one of the following graphs represents the curve?

$\frac{\sqrt{2} \cos 45^{\circ}+\cos 56^{\circ}+\cos 58^{\circ}-\cos 66^{\circ}}{\sqrt{2} \cos 28^{\circ} \cos 29^{\circ} \sin 33^{\circ}} = ?$

$\sec {50^o} + \tan {50^o}$ is equal to

The number of real numbers $\lambda$ for which the equality $\frac{\sin (\lambda \alpha) \cos (\lambda \alpha)}{\sin \alpha \cos \alpha} = \lambda - 1$ holds for all real $\alpha$ which are not integral multiples of $\pi/2$ is:

If $\operatorname{cosec}^2(\alpha+\beta)-\sin^2(\beta-\alpha)+\sin^2(2\alpha-\beta)=\cos^2(\alpha-\beta)$ where $\alpha, \beta \in (0, \frac{\pi}{2})$,then $\sin(\alpha-\beta)$ is equal to