If $\sec \theta = m$ and $\tan \theta = n$,then $\frac{1}{m} \left[ m + n + \frac{1}{m + n} \right] = $

If the orthocentre and circumcentre of a triangle $ABC$ are at equal distances from the side $BC$ and lie on the same side of $BC$,then the value of $\tan B \tan C$ is equal to:

If $\frac{\cos (A+B)}{\cos (A-B)}=\frac{\sin (C+D)}{\sin (C-D)}$,then $\tan A \tan B \tan C=$

Let $\alpha = 3 + 4 + 8 + 9 + 13 + 14 + \dots$ up to $40$ terms. If $(\tan \beta)^{1020}$ is a root of the equation $x^2 + x - 2 = 0$, where $\beta \in (0, \frac{\pi}{2})$, then $\sin^2 \beta + 3 \cos^2 \beta$ is equal to:

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

If $\cos (x-y), \cos x, \cos (x+y)$ are three distinct numbers which are in harmonic progression and $\cos x \neq \cos y$,then $1+\cos y$ is equal to