In a $\triangle ABC$,if $b=10$,$a \cos^2 \frac{C}{2} + c \cos^2 \frac{A}{2} = 15$,and the area of the triangle is $15\sqrt{3}$ sq. units,then $\cot \frac{B}{2} =$

In $\Delta ABC$,$a \cot A + b \cot B + c \cot C = . . . $ (where $r$ is inradius and $R$ is circumradius.)

In $\triangle ABC$,$(a-b)^2 \sin^2\left(\frac{A+B}{2}\right) + (a+b)^2 \sin^2\left(\frac{C}{2}\right) = $

If $2 \tan ^2 \theta-5 \sec \theta=1$ has exactly $7$ solutions in the interval $\left[0, \frac{n \pi}{2}\right]$,for the least value of $n \in N$,then $\sum_{k=1}^{n} \frac{k}{2^{k}}$ is equal to :

The sides $a, b, c$ of a triangle satisfy the relations $c^2=2ab$ and $a^2+c^2=3b^2$. Then the measure of $\angle BAC$,in degrees,is

Statement $-1$: The number of common solutions of the trigonometric equations $2\sin^2\theta - \cos 2\theta = 0$ and $2\cos^2\theta - 3\sin\theta = 0$ in the interval $[0, 2\pi]$ is two.
Statement $-2$: The number of solutions of the equation $2\cos^2\theta - 3\sin\theta = 0$ in the interval $[0, \pi]$ is two.