Show that `r_(1)+r_(2)=c cot ((C)/(2))`

The distance between two particles of masses m_(1)andm_(2) is r. If the distance of these particles from the centre of mass of the system are r_(1)andr_(2) respectively, then show that r_(1)=r((m_(2))/(m_(1)+m_(2)))andr_(2)=r((m_(1))/(m_(1)+m_(2)))

If A , B and C are interior angles of a triangle ABC, then show that tan ((A+B) /(2)) =cot (C/2)

Show that ((1+tan ^(2) A) /(1+cot ^(2) A ) ) =((1- tanA )/( 1-cot A ))^(2)= tan ^(2) A

Show that (1- tan ^(2) A ) /( cot ^(2) A-1) =tan ^(2) A

Show that (b-c)/(r _(1))+ (c-a)/(r _(2))+(a-b)/(r _(3)) =0.

Statement I In a Delta ABC, if a lt b lt c and ri si inradius and r_(1), r_(2) ,r_(3) are the exradii opposite to angle A,B,C respectively, then r lt r_(1) lt r_(2) lt r_(3). Statement II For, DeltaABC r_(1)r_(2)+r_(2)r_(3)+r_(3)r_(1)=(r_(1)r_(2)r_(3))/(r)

If A+B+C=pi , prove that : cot (A/2)+ cot(B/2) + cot(C/2) = cot(A/2) cot (B/2) cot (C/2)

Show that (d)/(dx) e^(ax) cos (bx + c) = r e^(ax) cos (bx + c + alpha) where r= sqrt(a^(2) + b^(2)), cosalpha= (a)/(r ), sin alpha = (b)/(r ) and (d^(2))/(dx^(2)) e^(ax) cos (ax + c) = r^(2) e^(ax ) cos (bx + c + 2 alpha) .

Prove that if two bubbles of radii r_(1) and r_(2)(r_(1)ltr_(2)) come in contact with each other then the radius of curvature of the common surface r=(r_(1)r_(2))/(r_(2)-r_(1))