In a right-angled triangle,the hypotenuse is $2\sqrt{2}$ times the length of the perpendicular drawn from the opposite vertex to the hypotenuse. Then,the other two angles are:

In $\triangle ABC$,find the value of $a^3 \cos(B-C) + b^3 \cos(C-A) + c^3 \cos(A-B)$.

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 :

In $\triangle ABC$,if $A = 60^{\circ}$ and $B = 105^{\circ}$,then find the value of $\frac{2R^2(b-c) \sin A \sin B \sin C}{(b+c)(s-a \cos C - c \cos A)(s-a \cos B - b \cos A)}$.

Suppose that the sides $a, b, c$ of a triangle $ABC$ satisfy $b^2 = ac$. Then the set of all possible values of $\frac{\sin A \cot C + \cos A}{\sin B \cot C + \cos B}$ is

Let $x, y$ and $z$ be positive real numbers. Suppose $x, y$ and $z$ are lengths of the sides of a triangle opposite to its angles $X, Y$ and $Z$,respectively. If $\tan \frac{X}{2} + \tan \frac{Z}{2} = \frac{2y}{x+y+z}$,then which of the following statements is/are $TRUE$?
$(A) 2Y = X + Z$
$(B) Y = X + Z$
$(C) \tan \frac{X}{2} = \frac{x}{y+z}$
$(D) x^2 + z^2 - y^2 = xz$