Sum to \(n\) terms \[ \frac{a}{a^2-1} + \frac{a^2}{a^4-1} + \frac{a^4}{a^8-1} + \dots \] and deduce the sum to \(n\) terms of \[ \operatorname{cosec}\theta + \operatorname{cosec}2\theta + \operatorname{cosec}2^2\theta + \dots. \]
If \(u_n - n(1+k)u_{n-1} + n(n-1)ku_{n-2}=0\), and \(u_2=2u_1k\), shew that \[ \frac{u_2}{2!}+\frac{u_3}{3!}+\frac{u_4}{4!}+\dots = u_1 e^k. \]
If \[ \sin^2\theta = \sin(A-\theta)\sin(B-\theta)\sin(C-\theta), \] and \[ A+B+C=\pi, \] prove that \[ \cot\theta = \cot A + \cot B + \cot C. \]
Two uniform rods \(AB, BC\) of equal weight are hinged at \(B\). The end \(A\) can turn about a fixed point and \(BC\) rests across a smooth horizontal peg. If in equilibrium both rods make angles of 60\(^{\circ}\) with the vertical, prove that the reaction at \(B\) divides the angle \(ABC\) into angles whose tangents are as \(-1:14\).
A circular disc can turn about a smooth pivot through its centre on a rough horizontal table. The pressure of the disc on the table is distributed uniformly. Shew that if \(\mu\) be the coefficient of friction and \(W\) the weight of the disc the least force that will turn the disc round the pivot is \(\frac{2}{3}\mu W\).
A perfectly elastic particle is dropped from a point on a fixed vertical circular hoop, shew that after two rebounds it will rise vertically if \[ 2\sin 4\theta = \tan\theta, \] where \(\theta\) is the angular distance of the point from the highest point of the hoop.
A particle of mass \(m\) is placed on a smooth wedge of mass \(M\) with one face vertical and the other inclined to the horizontal at an angle \(\alpha\) resting on a smooth horizontal table. If the wedge moves parallel to its lines of greatest slope and encounters a fixed obstacle which brings it to rest at the instant when its velocity is \(V\), shew that the magnitude of the impulse between the wedge and the obstacle is \(V(M+m\sin^2\alpha)\), and find the impulse between the wedge and the particle. Shew also that the velocity of the particle relative to the wedge is reduced instantaneously in the ratio \(M+m\sin^2\alpha:M+m\).
Determine \(\sin\frac{\pi}{10}\) and \(\sin\frac{\pi}{5}\), and prove that \[ 8\sin\frac{\pi}{10}\sin\frac{3\pi}{10} = \sqrt{(30-6\sqrt{5})} - \sqrt{5}-1. \]
Two flag-staffs of heights \(a\) and \(b\) stand on level ground at points \(A\) and \(B\). At a point \(P\) on the ground directly between \(A\) and \(B\) they subtend equal angles \(\alpha\) and at another point \(Q\) on the ground distant \(c\) from \(P\) they subtend equal angles \(\beta\). \(PQ\) makes with \(PA\) an angle \(\gamma\). Prove that \[ \frac{1}{a}-\frac{1}{b} = \frac{2}{c}\cot\alpha\cos\gamma, \] also that \[ \cot^2\beta - \cot^2\alpha = c^2/ab. \]
Prove that, when \(n\) is an even integer, \[ \cos n\theta = 1 - \frac{n^2}{2!}\sin^2\theta + \frac{n^2(n^2-2^2)}{4!}\sin^4\theta - \frac{n^2(n^2-2^2)(n^2-4^2)}{6!}\sin^6\theta - \dots. \] Prove that \[ \operatorname{cosec}^2\frac{\pi}{20} + \operatorname{cosec}^2\frac{3\pi}{20} + \operatorname{cosec}^2\frac{7\pi}{20} + \operatorname{cosec}^2\frac{9\pi}{20} = 48. \]