One root of the equation $\cos x - x + \frac{1}{2} = 0$ lies in the interval

If $\alpha_1, \alpha_2, \cdots, \alpha_n$ are in $A$.$P$. with common difference $\theta$, then the sum of the series $\sec \alpha_1 \sec \alpha_2 + \sec \alpha_2 \sec \alpha_3 + \cdots + \sec \alpha_{n-1} \sec \alpha_n = k(\tan \alpha_n - \tan \alpha_1)$, where $k=$

If $x + y = 3 - \cos 4\theta$ and $x - y = 4 \sin 2\theta$,then:

If $(1 + \sin A)(1 + \sin B)(1 + \sin C) = (1 - \sin A)(1 - \sin B)(1 - \sin C)$,then each side is equal to

$\sinh ^{-1} 2 + \sinh ^{-1} 3 = x \Rightarrow \cosh x$ is equal to

Let $\tan \alpha, \tan \beta$ and $\tan \gamma$ (where $\alpha, \beta, \gamma \neq \frac{(2n-1)\pi}{2}, n \in N$) be the slopes of three line segments $OA, OB$ and $OC$ respectively,where $O$ is the origin. If the circumcentre of $\Delta ABC$ coincides with the origin and its orthocentre lies on the $y$-axis,then the value of $\left(\frac{\cos 3\alpha + \cos 3\beta + \cos 3\gamma}{\cos \alpha \cos \beta \cos \gamma}\right)^2$ is equal to: