Find the \(n\) real quadratic factors of \(x^{2n}-2a^nx^n\cos n\phi+a^{2n}\). Show that \(\prod_{r=0}^{r=n-1} \left\{\cos\phi-\cos\frac{2r\pi}{n}\right\} + \prod_{r=0}^{r=n-1} \left\{1-\cos\left(\phi+\frac{2r\pi}{n}\right)\right\} = 0\).
Show that the function \(\sin x + a\sin 3x\) for values of \(x\) between \(0\) and \(\pi\) has two minima with a maximum between, if \(a < -\frac{1}{3}\); one maximum, if \(-\frac{1}{3} < a < \frac{1}{3}\); two maxima with a minimum between, if \(a > \frac{1}{3}\).
Show how to find the asymptotes of an algebraic curve without discussing exceptional cases. Find the asymptotes of the curve \(x^2y+xy^2+xy+y^2+3x=0\). Trace the curve.
Find the values of \(\int \sec x dx, \int x^n\log x dx, \int \frac{dx}{x\sqrt{a^2+x^2}}\). Show that \[ \int_b^a \frac{dx}{x\sqrt{(a-x)(x-b)}} = \frac{\pi}{\sqrt{ab}}, \quad (a>b>0), \] and that \[ \int_0^1 x^4 \sqrt{1-x^2} dx = \frac{5\pi}{256}. \]
If \(u_{p,q}=\int_0^{\pi/2}(\cos x)^p\cos qx dx\), prove the reduction formulae \[ u_{p,q} = \frac{p(p-1)}{p^2-q^2}u_{p-2,q} = \frac{p}{p+q}u_{p-1,q-1}. \]
Prove geometrically that \(\tan A = \frac{\sin 2A}{1+\cos 2A}\). If \(ABC\) is a triangle, prove that \[ \sin 3A \cos A + \sin 3B \cos B + \sin 3C \cos C = 2\sin A \sin B \sin C (3+2\cos 2A+2\cos 2B+2\cos 2C). \]
If \(P\) is the orthocentre of a triangle \(ABC\), \(O\) the centre of the circumscribing circle, and \(R\) the length of its radius, prove that \[ OP^2 = R^2(1-8\cos A\cos B\cos C). \] Prove also that if \(Q\) is the middle point of \(OP\), \[ AQ^2+BQ^2+CQ^2=3R^2-\frac{1}{4}OP^2. \]
Express \(x^{2n}-2x^n\cos n\theta+1\) as the product of \(n\) real quadratic factors and deduce that \[ \cos n\theta = 2^{n-1} \sin\left(\theta+\frac{\pi}{2n}\right) \sin\left(\theta+\frac{3\pi}{2n}\right)\dots\sin\left(\theta+\frac{2n-1}{2n}\pi\right). \] Prove that \[ \cos\frac{2\pi}{7}+\cos\frac{4\pi}{7}+\cos\frac{6\pi}{7} = -\frac{1}{4}. \]
Expand \(\cos x\) in ascending powers of \(x\), and prove that \[ \cos x \cosh x = 1 - \frac{2^2x^4}{4!} + \frac{2^4x^8}{8!} - \dots. \]
Find the equations of the bisectors of the angles between the straight lines \[ ax+by=c \quad \text{and} \quad bx+ay=d. \] Find the coordinates of the centre of the inscribed circle of the triangle the equations of whose sides are \[ x+y=1, \quad x-y=3 \quad \text{and} \quad 17x+7y+3=0. \]