If $A(n) = \sin^n \alpha + \cos^n \alpha$,then $A(1) A(4) + A(2) A(5) =$

Let $A = \{x \in R : |\sqrt{3} \cos x - \sin x| \geq 2, 0 \leq x \leq 2\pi\}$. If $x_1 \in A$ and $x_2 \in A$,then find the possible value of $\frac{x_1}{x_2}$.

If $\sin \theta + \sin \phi = a$ and $\cos \theta + \cos \phi = b,$ then $\tan \frac{\theta - \phi}{2}$ is equal to

Let $|\cos \theta \cos (60^{\circ}-\theta) \cos (60^{\circ}+\theta)| \leq \frac{1}{8}$, where $\theta \in [0, 2\pi]$. Then, the sum of all $\theta \in [0, 2\pi]$ where $\cos 3\theta$ attains its maximum value is: (in $\pi$)

Let $a, b, c$ be three non-zero real numbers such that the equation $\sqrt{3} a \cos x + 2 b \sin x = c$,$x \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ has two distinct real roots $\alpha$ and $\beta$ with $\alpha + \beta = \frac{\pi}{3}$. Then the value of $\frac{b}{a}$ is

If $\cos \theta - 4 \sin \theta = 1$,then $\sin \theta + 4 \cos \theta$ is equal to