Resolve \[ \frac{1}{(1-x)^2(1+x^2)} \] into partial fractions. Prove that, if this function is expanded in a series of ascending powers of \(x\), the coefficient of \(x^{4n}\) is \(2n+1\) and each of the coefficients of \(x^{4n-3}, x^{4n-2}\), and \(x^{4n-1}\) is \(2n\).
Form the equation whose roots are the sum and product of the reciprocals of the roots of the equation \[ x^2 + \lambda x + \mu = 0. \] If the equation thus formed is identical with the original quadratic equation, prove that \[ \lambda^2 = (1 - \lambda)^2 \] and \[ \mu^3 = \mu^2 + 1. \]
Prove that the \(n\)th convergents of the two continued fractions \[ \frac{1}{a+} \frac{1}{b+} \frac{1}{a+} \frac{1}{b+} \frac{1}{a+} \frac{1}{b+} \dots, \] \[ \frac{a}{c} \left( \frac{1}{c+} \frac{1}{c+} \frac{1}{c+} \dots \right) \quad \text{where } c = \sqrt{(ab)}, \] are equal. Hence shew that, if \(x\) is any convergent to the first continued fraction, the succeeding convergent is \(a + \frac{a}{bx}\).
The internal bisectors of the angles \(A, B, C\) of a triangle meet the circumcircle in \(A', B', C'\). Prove that, if \(\Delta, \Delta'\) are the areas of the triangles \(ABC, A'B'C'\), \[ \frac{\Delta'}{\Delta} = \frac{R}{2r}. \] Prove also that, if \(B'C'\) meets \(AB, AC\) in \(Z\) and \(Y\), \[ AY = AZ = 2Rr/a, \] and from this and similar results shew that the diagonals of the convex hexagon formed by the sides of the triangles \(ABC, A'B'C'\) pass through the incentre of the triangle \(ABC\).
A vertical flagstaff \(AB\) is observed to subtend the same angle at two points \(P, Q\) at the same level in a plane through \(AB\). The elevations of the base of \(AB\) as seen from \(P, Q\) are \(\alpha, \beta\) (\(\alpha > \beta\)) and the distance between \(P\) and \(Q\) is \(a\). Prove that \[ AB = a \cos(\alpha + \beta)/\sin(\alpha - \beta). \]
Find all the values of \(\theta\) that satisfy the equation \[ \tan\theta \cot(\theta+\alpha) = \tan\beta \cot(\beta+\alpha). \] Eliminate \(\theta\) from \[ a \cos(\alpha - 3\theta) = 2b \cos^3\theta \quad \text{and} \quad a \sin(\alpha - 3\theta) = 2b \sin^3\theta. \]
Prove that \(\cos n\theta\) where \(n\) is an integer can be expressed as a rational function of \(\cos\theta\) of degree \(n\). Prove that, if \(2 \cos\theta = x\), \[ \frac{1+\cos 9\theta}{1+\cos\theta} = (x^4 - x^3 - 3x^2 + 2x + 1)^2. \]
In a triangle prove that
Expand \(\log(1+2h\cos\theta+h^2)\) in the form \(\sum A_n h^n \cos n\theta\) and find \(A_n\). If \[ y = \log\left\{\tan\left(\frac{\pi}{4}+\frac{x}{2}\right)\right\} = a_1 x + a_3 x^3 + a_5 x^5 + \dots, \] prove that \[ x = a_1 y - a_3 y^3 + a_5 y^5 - \dots . \]
Prove that in general any system of coplanar forces can be reduced to a single force acting through a given point together with a couple. A triangle \(ABC\) is formed of three light rods freely jointed to each other at their ends. The rods are acted on by coplanar forces at their middle points perpendicularly to the rods and respectively proportional to the rods. Prove that if the forces all act outwards the system is in equilibrium and that the stresses at the joints are all equal and act along the tangents at the angular points to the circumcircle of the triangle \(ABC\).