In a $\Delta ABC$,if $\sin A + \sin B + \sin C = 1 + \sqrt{2}$ and $\cos A + \cos B + \cos C = \sqrt{2}$,then the triangle is:

If $A, B, C$ are the angles of a triangle,then $\sum \frac{\cot A + \cot B}{\tan A + \tan B} = $

In a triangle $ABC$,$AD$ is the altitude from $A$. Given $b > c$,$\angle C = 23^o$ and $AD = \frac{abc}{b^2 - c^2}$,then $\angle B = \dots^o$.

Let $PQRS$ be a quadrilateral in a plane,where $QR = 1$,$\angle PQR = \angle QRS = 70^{\circ}$,$\angle PQS = 15^{\circ}$ and $\angle PRS = 40^{\circ}$. If $\angle RPS = \theta^{\circ}$,$PQ = \alpha$ and $PS = \beta$,then the interval$(s)$ that contain$(s)$ the value of $4 \alpha \beta \sin \theta^{\circ}$ is/are
$(A)$ $(0, \sqrt{2})$
$(B)$ $(1, 2)$
$(C)$ $(\sqrt{2}, 3)$
$(D)$ $(2 \sqrt{2}, 3 \sqrt{2})$

In $\triangle PQR$,if $\angle R = \frac{\pi}{4}$ and $\tan(\frac{P}{3})$,$\tan(\frac{Q}{3})$ are the roots of the equation $ax^2 + bx + c = 0$,then:

If in a triangle $ABC,$ $\cos A \cos B + \sin A \sin B \sin C = 1,$ then the sides are proportional to