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

The angles of a triangle are in $A.P.$ and the greatest angle is double the least angle. Find the sine of the third angle.

In a triangle $ABC$,if $a < b < c$ and $\frac{a^3+b^3+c^3}{\sin^3 A+\sin^3 B+\sin^3 C}=8$,then the maximum value of $c$ is

The angles of depression of the top and the foot of a chimney as seen from the top of a second chimney,which is $150 \, m$ high and standing on the same level as the first,are $\theta$ and $\phi$ respectively. If $\tan \theta = \frac{4}{3}$ and $\tan \phi = \frac{5}{2}$,then the distance between their tops is.......$m$.

If $\alpha$ and $\beta$ are different values of $x$ satisfying $a \cos x + b \sin x = c,$ then $\tan \left( \frac{\alpha + \beta}{2} \right) = $

$AB$ is a vertical tower. The point $A$ is on the ground and $C$ is the middle point of $AB$. The part $CB$ subtends an angle $\alpha$ at a point $P$ on the ground. If $AP = n \cdot AB$,then the correct relation is: