Show how to express \(x^n + \frac{1}{x^n}\) in terms of \(x+\frac{1}{x}\). Obtain the roots of the equation \(x^4 - 6x^3 + 13x^2 - 6x + 1 = 0\) in the form \((\sqrt{3} \pm \sqrt{2})e^{\pm i\frac{\pi}{6}}\).
If \[ \frac{1}{(x+1^2)(x+2^2)\dots(x+n^2)} = \frac{A_1}{x+1^2} + \frac{A_2}{x+2^2} + \dots + \frac{A_n}{x+n^2}, \] show that \[ A_r = \frac{(-1)^{r-1} 2r^2}{(n-r)!(n+r)!}, \] where \(0!\) is taken to mean unity. Hence show that if \(p\) is any integer \(< n\), \[ \frac{1^{2p}}{(n-1)!(n+1)!} - \frac{2^{2p}}{(n-2)!(n+2)!} + \dots + \frac{(-1)^{n-1}n^{2p}}{(2n)!} = 0. \]
Examine the convergence of the series whose \(n\)th term is \(\frac{x^n}{x^{2n}+x^n+1}\) for any value of \(x\). Find the sum of the series \(\sum_{n=2}^\infty \frac{n^3 x^n}{n^4-1}\), when convergent.
Prove that for the continued fraction \(a_1 + \frac{1}{a_2+} \frac{1}{a_3+} \dots\) where the \(a\)'s are all positive, any convergent is intermediate in magnitude between the next two preceding ones. For the fraction \(a+\frac{1}{b+}\frac{1}{a+}\frac{1}{b+}\frac{1}{a+}\frac{1}{b+}\dots\), prove that if \(p_n/q_n\) is the \(n\)th convergent, \(p_{2n}=q_{2n+1}\) and \(bp_{2n-1}=aq_{2n}\).
Find all the values of \(x\) which satisfy the equation \(\cos 3x = \cos 3a\), where \(a\) is given; and prove their existence geometrically. Solve the equation \(\cos x \cos c + \sin a \sin b = \cos(x-a)\cos(x-b)\), where \(a, b, c\) are given.
In any triangle prove that \[ r=4R\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}. \] If a point D in the side BC of a triangle ABC be such that the inscribed circles of the triangles ADB, ADC are equal, show that AD meets BC at an angle \(\theta\) given by \(a \cos \theta = b \sim c\).
Show how to obtain all the \(n\)th roots of \(a+ib\), where \(a, b\) are real. If the roots of \(t^2-2t+2=0\) are \(\alpha, \beta\), show that \[ \frac{(x+\alpha)^n - (x+\beta)^n}{\alpha-\beta} = \frac{\sin n\phi}{\sin^n\phi}, \text{ where } x+1=\cot\phi. \]
Find the \(n\)th differential coefficient of (i) \(e^{ax}\cos bx\), (ii) \(\frac{\log x}{x}\). Prove that \[ \cosh ax \cos bx = 1 + \frac{x^2c^2\cos 2\alpha}{2!} + \dots + \frac{x^{2n}c^{2n}\cos 2n\alpha}{(2n)!} + \dots, \] where \(\tan\alpha = \frac{b}{a}\) and \(c^2=a^2+b^2\).
State sufficient conditions for \(f(x)\) to be a maximum when \(x=a\). Show that the angle \(\phi\) between the radius vector and tangent to a curve is a maximum or minimum at any point where the radius of curvature \(\rho\) subtends a right angle at the pole; show also that, omitting the case where the curve passes through the pole, \(\phi\) is a maximum if \(\frac{d\rho}{ds} - \cot\phi > 0\) at the point.
Prove the formula \(\rho = p + \frac{d^2p}{d\psi^2}\) for a plane curve. For the curve \begin{align*} x &= a(n\cos t + \cos nt), \\ y &= a(n\sin t + \sin nt), \end{align*} show that \(p = a(n+1)\cos\left(\frac{n-1}{n+1}\psi\right)\), where \(p\) is the length of the perpendicular from the origin on the tangent, and \(\psi\) is the inclination of \(p\) to the axis of \(x\). Show that if \(n\) is a positive integer, the curve has \((n-1)\) cusps.