If $\frac{\cos 3A + \sin 3A}{\cos A - \sin A} = 1 - K \sin 2A,$ the value of $K$ is

The value of $(\sec \theta + \operatorname{cosec} \theta)$ when $\theta = 45^{\circ}$ is: (in $\sqrt{2}$)

Let $f(x) = \max (\sin x, \cos x)$ and $g(x) = \min (\cos x, \sin x)$. If the equation $f(x)y^2 + ay + g(x) = 0$ has real roots for all $x \in R$,then the complete set of values of $a$ is:

If $\tan A + \sin A = m$ and $\tan A - \sin A = n,$ then $\frac{(m^2 - n^2)^2}{mn} = $

The value of $\frac{1}{4} \tan \frac{\pi}{8} + \frac{1}{8} \tan \frac{\pi}{16} + \frac{1}{16} \tan \frac{\pi}{32} + \dots \infty$ terms is equal to-

$ABC$ is a triangle. If $\sin \left(\frac{A+B}{2}\right) = \frac{\sqrt{3}}{2}$,then the value of $\sin \frac{C}{2}$ is