In $\triangle ABC$,if $\cos A \cos B + \sin A \sin B \sin C = 1$ and $C = \frac{\pi}{2}$,then $A : B =$

Let $ABC$ be a triangle and let $D$ be the mid-point of $BC$. Suppose $\cot (\angle CAD) : \cot (\angle BAD) = 2 : 1$. If $G$ is the centroid of $\triangle ABC$,then the measure of $\angle BGA$ is (in $^{\circ}$)

In $\triangle ABC$,right-angled at $A$,the circumradius,inradius,and radius of the excircle opposite to $A$ are respectively in the ratio $2:5:\lambda$. Then the roots of the equation $x^2-(\lambda-5)x+(\lambda-6)=0$ are:

$A$ tower $AB$ leans towards west making an angle $\alpha$ with the vertical. The angular elevation of $B$,the top most point of the tower,is $\beta$ as observed from a point $C$ due east of $A$ at a distance $d$ from $A$. If the angular elevation of $B$ from a point $D$ due east of $C$ at a distance $2d$ from $C$ is $\gamma$,then $2\tan \alpha$ can be given as

In triangle $ABC$,$\frac{\tan A}{2} = \frac{\tan B}{3} = \frac{\tan C}{4}$,then the value of $\sec^2 A + \sec^2 B + \sec^2 C$ is:

If $ABCD$ is a cyclic quadrilateral,then the value of $\cos A - \cos B + \cos C - \cos D = $