Prove that \begin{align*} 1 - \cos^2\theta - \cos^2\phi &- \cos^2\psi + 2\cos\theta\cos\phi\cos\psi \\ &= 4\sin\sigma\sin(\sigma-\theta)\sin(\sigma-\phi)\sin(\sigma-\psi), \end{align*} where \(2\sigma = \theta+\phi+\psi\). If \(x\) and \(y\) satisfy the equation \[ \sin x + \sin y = \sqrt{3}(\cos y - \cos x), \] prove that \(\sin 3x + \sin 3y = 0\).
In any triangle, prove that
Prove that \begin{align*} \sin n\theta/\sin\theta = 2^{n-1}\cos^{n-1}\theta &- \frac{n-2}{1}2^{n-3}\cos^{n-3}\theta \\ &+ \frac{(n-3)(n-4)}{2!}2^{n-5}\cos^{n-5}\theta - \dots \end{align*} and find the general term. Prove that \(\prod_{r=1}^{r=5} \cos \frac{r\pi}{11} = \frac{1}{2^5}\).
Determine the conditions that a system of coplanar forces acting at a point should be in equilibrium. Four forces act along the sides of a quadrilateral \(ABCD\) and are represented in direction and magnitude by \(BA, BC, AD, CD\). Prove that their resultant is parallel to one diagonal and bisects the other.
State the laws of friction. A uniform rod lying on a rough inclined plane can rotate about a point on it at distances \(a\) and \(b\) from its ends; shew that if \(\mu\) be the coefficient of friction, and \(\alpha\) the angle of the plane, the inclination of the rod to the line of greatest slope cannot be greater than \[ \sin^{-1}\left[\mu\cot\alpha\sqrt{\frac{a^2+b^2}{a^2\sim b^2}}\right]. \]
State the Principle of Virtual Work and shew how it can be applied to find the stress in a rod of a jointed framework. A triangle \(ABC\) of any shape is formed of light rods smoothly jointed to each other at their ends. It is placed in a vertical plane with \(A\) downwards and the rods \(AB, AC\) resting on two smooth pegs in a horizontal line. A weight \(W\) is suspended from \(A\); prove that the stress in the rod \(BC\) is \[ \frac{1}{2}\frac{Wl}{p}\text{cosec}^2 A, \] where \(2l\) is the distance between the pegs and \(p\) is the perpendicular from \(A\) on \(BC\).
Define angular velocity. A point is describing a circle with uniform velocity; prove that the angular velocity of the line joining it to any point on the circumference of the circle is constant. One end \(A\) of the connecting rod of a piston is moving in a circle with velocity \(u\) while the other end \(B\) is moving in a straight line which passes through \(O\) the centre of the circle with velocity \(v\). Prove that \(v=u\sin BAO \sec OBA\), and find the angular velocity of \(AB\).
A string passing over a smooth fixed pulley carries a mass \(2m\) at one end and another smooth pulley of mass \(m\) at the other end. A string having masses \(m\) and \(2m\) at its ends passes over the second pulley. If motion of the system takes place under gravity, find the accelerations of the various parts of the system and the acceleration with which the centre of gravity of the whole system descends.
A point is moving in a circle with velocity \(v\). Prove that \(v^2/r\) is its acceleration towards the centre. A heavy particle is suspended from a point by a string of length \(a\). Prove that if it is projected when at its lowest point with a horizontal velocity greater than \(\sqrt{5ag}\) it will describe a circle in the vertical plane. Also describe the motion of the particle when its velocity of projection (i) lies between \(\sqrt{5ag}\) and \(\sqrt{2ag}\), (ii) is less than \(\sqrt{2ag}\).
Given the inscribed and circumscribed circles of a triangle in position, prove that the orthocentre lies on a fixed circle.