In a triangle $ABC,$ ${a^3}\cos (B - C) + {b^3}\cos (C - A) + {c^3}\cos (A - B) = $

In a right-angled triangle,if the difference between the two acute angles is $60^{\circ}$,then the ratio of the length of the hypotenuse to the length of the perpendicular drawn to the hypotenuse from its opposite vertex is: (in $: 1$)

If the angles $A, B$ and $C$ of a triangle are in an Arithmetic Progression and if $a, b$ and $c$ denote the lengths of the sides opposite to $A, B$ and $C$ respectively,then the value of the expression $\frac{a}{c} \sin 2C + \frac{c}{a} \sin 2A$ is

Let $p_1, p_2, p_3$ be the altitudes of a triangle $ABC$ drawn through the vertices $A, B, C$ respectively. If $r_1=4, r_2=6, r_3=12$ are the ex-radii of triangle $ABC$,then $\frac{1}{p_1^2}+\frac{1}{p_2^2}+\frac{1}{p_3^2}=$

In $\triangle ABC$,if $3 \sin A + 4 \cos B = 6$ and $4 \sin B + 3 \cos A = 1$,then the $\angle C$ 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