Problem source

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\rput[tl](-2.51,-0.16){$m$}
\rput[tl](0.61,-0.16){$km$}
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A particle $P_1$ of mass $m$ collides with  a particle $P_2$
of mass $km$ which is at rest. No energy is lost in the collision.
 The direction of motion 
of $P_1$ and $P_2$ after the collision make 
non-zero
angles of  $\theta$ and $\phi$, respectively, with the direction of motion 
of $P_1$ before the collision, as shown. Show that
\[
\sin^2\theta + k\sin^2\phi = k\sin^2(\theta+\phi) \;.
\] 
Show that, if the angle between the particles after the collision is a right angle,
then $k=1\,$.