In a triangle $ABC$,if $A = \frac{\pi}{4}$ and $\tan B \tan C = K$,then $K$ must satisfy:

In a triangle $ABC$,with usual notations,$\frac{\cos B+\cos C}{b+c}+\frac{\cos A}{a}$ has the value

Let $S=\{x \in(-\pi, \pi) \mid x \neq 0, \pm \frac{\pi}{2}\}$. The sum of all distinct solutions of the equation $\sqrt{3} \sec x+\operatorname{cosec} x+2(\tan x-\cot x)=0$ in the set $S$ is equal to

In $\triangle ABC$,if $\cos A + \cos C = 4 \sin^2 \frac{B}{2}$,then the ratio between the perimeter of the triangle and $(a+c)$ is

Statement-$I$: In the interval $[0, 2\pi]$,the number of common solutions of the equations $2 \sin^2 \theta - \cos 2\theta = 0$ and $2 \cos^2 \theta - 3 \sin \theta = 0$ is two.
Statement-$II$: The number of solutions of $2 \cos^2 \theta - 3 \sin \theta = 0$ in $[0, \pi]$ is two.

Let $p, q$ and $r$ be the altitudes of a triangle with area $S$ and perimeter $2t$. Then, the value of $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}$ is