Plane trigonometry, by S.L. Loney.

480 TRIGONOMETRY. [Misc. Exs. LXIX.J
23. Prove that the general value of sinh-lx is
ik7r + ( - 1)k log [x + /14 + x2],
where k is any integer.
24. If Pi, P2...... pn be the distances of the vertices of a regular
polygon of n sides from any point P in its plane, prove that
i I I n 212n- a~n
1 1 1 n.2n-_ a2n
12 PL ~2 = r4,2 a2 2n - 2anrn cos nO + a2n
where a is the radius of the circumcircle of the polygon, 9r is the distance
of P from its centre 0, and 0 is the angle that OP makes with the radius
to any angular point of the polygon.
25. If O+,q+ -=27r, prove that
Cos2 0 + COS2 q + cOS2 p - 2 cos O cos q cos '= 1.
Hence deduce the relation between the six straight lines joining
4 points which are in one plane.