In a triangle $ABC$,angle $A$ is greater than angle $B$. If the measures of angles $A$ and $B$ satisfy the equation $3\sin x - 4\sin^3 x - k = 0$ for $0 < k < 1$,then the measure of angle $C$ is:

If $\tan (\pi \cos \theta)=\cot (\pi \sin \theta)$,then a value of $\cos \left(\theta-\frac{\pi}{4}\right)$ among the following 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 $TRUE$?
$(A) 2Y = X + Z$
$(B) Y = X + Z$
$(C) \tan \frac{X}{2} = \frac{x}{y+z}$
$(D) x^2 + z^2 - y^2 = xz$

In a $\triangle ABC$,if $A-B=120^{\circ}$ and $R=8r$,then find the value of $\frac{1+\cos C}{1-\cos C}$.

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

In $\triangle ABC$, with usual notations, if the angles are in Arithmetic Progression and $3c^2 = 2b^2$, then the measure of angle $A$ is