$A$ circular wire of diameter $10\,cm$ is cut and placed along the circumference of a circle of diameter $1\,m$. The angle subtended by the wire at the centre of the circle is equal to

If $\sin 3A = \cos (A - 26^{\circ})$,where $3A$ is an acute angle,then the value of $A$ is (in $^{\circ}$)

Let $f(x) = Ax^3 - Bx - \tan x \cdot \text{sgn}(x)$ be an even function for all $x \in \mathbb{R} - \left\{ (2n + 1) \frac{\pi}{2}, n \in \mathbb{Z} \right\}$, where $A = \sin^2 \alpha - \sin \alpha + \frac{1}{4}$ and $B = \tan^2 \alpha + \frac{2}{\sqrt{3}} \tan \alpha + \frac{1}{3}$. Then the number of values of $\alpha$ in $\left[ -\frac{3\pi}{2}, 2\pi \right]$ is (where $\text{sgn}(x)$ denotes the signum function of $x$).

Suppose $\theta$ and $\phi (\neq 0)$ are such that $\sec(\theta + \phi)$,$\sec\theta$,and $\sec(\theta - \phi)$ are in $A.P.$ If $\cos\theta = k \cos(\frac{\phi}{2})$ for some $k$,then $k$ is equal to

If $\cos(\alpha + \beta) = \frac{4}{5}$ and $\sin(\alpha - \beta) = \frac{5}{13}$,where $0 \le \alpha, \beta \le \frac{\pi}{4}$. Then $\tan 2\alpha = $

If $\sin \theta + \csc \theta = 2$,then $\sin^2 \theta + \csc^2 \theta = $