If \[ (1+3\sin^2\phi)^{\frac{1}{3}} = \sin^{\frac{2}{3}}\theta + \cos^{\frac{2}{3}}\theta, \] prove that \[ (1+3\tan^2\phi)\tan\theta = 2\tan^3\phi. \]
An attempt is made to construct a right angle by means of three strings of lengths 3, 4 and 5 yards. If the third string is one inch too long, find the resulting error in the right angle in minutes of arc.
Expand \(\frac{\sin n\theta}{\sin\theta}\) in a series of descending powers of \(\cos\theta\), when \(n\) is an odd integer. \par Prove also that when \(n\) is odd \[ n\operatorname{cosec} n\theta = \operatorname{cosec}\theta + \operatorname{cosec}\left(\theta+\frac{2\pi}{n}\right) + \operatorname{cosec}\left(\theta+\frac{4\pi}{n}\right) + \dots \text{ to } n \text{ terms}. \]
Prove that in areal coordinates the equation of the circumcircle of the triangle of reference is \(a^2yz+b^2zx+c^2xy=0\), and that the polar reciprocal of this circle with respect to the conic \(ux^2+vy^2+wz^2=0\) is \[ a\sqrt{(ux)}+b\sqrt{(vy)}+c\sqrt{(wz)}=0. \]
Find from the definition the derivative of \(\sin^{-1}x\). \par Prove that for the value \(x=0\), \(\frac{d^n}{dx^n}(\sin^{-1}x)=(1,3,5,\dots n-2)^2\) or 0 according as \(n\) is odd or even.
Find the maximum and minimum values of \(y=(x+1)^2(x+3)^3(x+2)\) and draw a rough graph of the curve.
Prove that the radius of curvature at any point of a curve is given by \[ \rho = \frac{(x'^2+y'^2)^{\frac{3}{2}}}{x'y''-y'x''}, \] where accents denote differentiations with regard to a parameter. \par Find the curvature at the origin of each of the branches of the curve \[ x^3-x^2y-xy^2+y^3=4xy-2y^2, \] and trace the curve.
If \(z=\frac{xy}{x-y}\), find all the second order differential coefficients of \(z\) with respect to \(x\) and \(y\), and verify that \[ x^2\frac{\partial^2 z}{\partial x^2} + 2xy\frac{\partial^2 z}{\partial x \partial y} + y^2\frac{\partial^2 z}{\partial y^2} = 0. \]
Obtain formulae of reduction for \[ \int x^n\cos mx\,dx, \quad \int x^k(a+bx^n)^p\,dx. \]
Find the area of the surface generated by the revolution of the lemniscate \(r^2=a^2\cos 2\theta\) round the initial line.