Prove that:
Express \(1-\cos^2\theta-\cos^2\phi-\cos^2\psi-2\cos\theta\cos\phi\cos\psi\) as a product of four cosines. Eliminate \(\theta\) from \(a\sin\theta+b\cos\theta=2c\cos2\theta\), \(a\cos\theta-b\sin\theta=c\sin2\theta\).
\(ABCD\) is a quadrilateral circumscribing a circle and \(a,b,c,d\) are the lengths of the tangents from \(A,B,C,D\) respectively; prove that the sum of a pair of opposite angles is \(2\theta\), where \((a+b)(b+c)(c+d)(d+a)\cos^2\theta=(ac-bd)^2\).
Express \((a+ib)^{c+id}\) in the form \(A+iB\) where \(i=\sqrt{-1}\). If \(\sin x = y\cos(x+a)\), expand \(x\) in ascending powers of \(y\).
State the laws of friction. On the radius \(OA\) of a circular disc as diameter a circle is described, and the disc enclosed by it is cut out. If the remaining solid rest in a vertical plane on two rough pegs in a horizontal plane subtending an angle \(2\alpha\) at the centre \(O\), show that the greatest angle that \(OA\) can make with the vertical is \(\sin^{-1}(3\sin2\lambda\sec\alpha)\), where \(\lambda\) is the angle of friction at the pegs.
State the principle of virtual work and prove it in the case of a single lamina acted on by forces in its plane. \(ABCD\) is a rhombus formed by four light rods smoothly jointed at their ends and \(PQ\) is a light rod smoothly jointed at one end to a point \(P\) in \(BC\) and at the other end to a point \(Q\) in \(AD\). Two forces each equal to \(F\) are applied at \(A\) and \(C\) in opposite directions along \(AC\). Prove that the stress in \(PQ\) is \(F.AB.PQ/AC(AQ\sim BP)\).
A projectile of mass \(m\) lb., moving horizontally with velocity \(v\) feet per second, strikes an inelastic nail of mass \(m'\) lb. projecting horizontally from a mass of \(M\) lb. which is free to slide on a smooth horizontal plane. Prove that the nail is driven \[ \frac{m^2M}{(M+m+m')(m+m')}\frac{6v^2}{gP} \text{ inches} \] into the block, where \(P\) lb. weight is the mean resistance of the block to penetration by the nail.
State the principle of the conservation of linear momentum. A smooth inclined plane of angle \(\alpha\) and of mass \(M\) is free to slide on a smooth horizontal plane. A particle of mass \(m\) is placed on its inclined face and slides down under gravity. Find its acceleration in space and the pressure between it and the plane.
Two equal light rods \(AB, BC\) are smoothly jointed at \(B\) and \(A\) is smoothly jointed to a fixed point. Masses \(m,m'\) are attached to \(B\) and \(C\) respectively. \(C\) is released from rest when \(AC\) is horizontal and when the angle \(ABC\) is \(2\pi/3\). Find the acceleration of \(C\) and the tension in \(AB\) immediately after the system is released.
Calculate the loss of kinetic energy when a ball of mass \(m\) moving with velocity \(u\) strikes directly a ball of mass \(m'\) moving with velocity \(u'\). Two equal balls are lying in contact on a smooth table, and a third equal ball, moving along their common tangent strikes them simultaneously. Prove that \(\frac{2}{3}(1-e^2)\) of its kinetic energy is lost by the impact, \(e\) being the coefficient of restitution for each pair of balls.