Problems

Establish the law of formation of successive convergents to a continued fraction. Prove that the product of the first \(n\) convergents to the fraction \[ \frac{12}{7-} \frac{12}{7-} \frac{12}{7-} \dots \text{ is } \frac{12^n}{4^{n+1}-3^{n+1}}. \]

If \(\alpha, \beta, \gamma, \delta\) are the angles of a plane quadrilateral, prove that \[ \cos 2\alpha + \cos 2\beta + \cos 2\gamma + \cos 2\delta = 4 \cos(\alpha+\beta) \cos(\alpha+\gamma) \cos(\alpha+\delta). \] Prove that the real solutions of the equation \[ \tan^2 x \tan \frac{x}{2} = 1 \] satisfy the equation \[ \cos 2x = 2-\sqrt{5}. \]

Prove the formula \[ \cos(A+B) = \cos A \cos B - \sin A \sin B \] for all real values of \(A\) and \(B\). If \[ \tan(\theta-\alpha)+\tan(\theta-\beta) = \tan(\phi-\alpha)+\tan(\phi-\beta) = 2\tan\gamma, \] and \((\theta-\phi)\) is not a multiple of \(\pi\), prove that \[ \alpha+\beta+\gamma-\theta-\phi \] is an odd multiple of \(\frac{\pi}{2}\).

In any triangle, prove that \[ r = 4R \sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}. \] A point \(D\) is taken in the side \(BC\) of a triangle \(ABC\), so that the inscribed circles of the triangles \(ADB\) and \(ADC\) are equal. Prove that the cosine of the angle \(ADB\) is \[ \pm\sin\frac{B-C}{2}\sec\frac{A}{2}. \]

Three vertical flagstaffs stand on a horizontal plane. At each of the points \(A, B\) and \(C\) in the horizontal plane, the tops of two of them are seen in the same straight line, and these straight lines make angles \(\alpha, \beta, \gamma\) with the horizontal. The plane containing the tops makes an angle \(\theta\) with the horizontal. Prove that the heights of the flagstaffs are \[ BC/\{\sqrt{\cot^2\beta - \cot^2\theta} + \sqrt{\cot^2\gamma - \cot^2\theta}\}, \] and two similar expressions.

Find \(n\) real factors of \(\cos n\theta - \cos n\alpha\). Sum to infinity the series

1. [(i)] \(\cos\theta + \frac{1}{2}\cos 2\theta + \frac{1}{3}\cos 3\theta + \dots\)
2. [(ii)] \(\sin\alpha + x\sin(\alpha+\beta) + \frac{x^2}{2!}\sin(\alpha+2\beta) + \dots + \frac{x^n}{n!}\sin(\alpha+n\beta) + \dots\)

Prove geometrically

1. [(i)] \(\tan\frac{A}{2} = \frac{\sin A}{1+\cos A}\),
2. [(ii)] \(\tan(45^\circ+A) + \tan(45^\circ-A) = 2\sec 2A\).

Shew that \[ \sum \cos^3 A \sin(B-C) = \cos(A+B+C) \sin(B-C)\sin(C-A)\sin(A-B). \]

In a triangle \(ABC\) prove that \(\frac{r}{R} = 4\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}\). Prove also that, if the bisectors of the angles meet the circumcircle in \(D, E, F\), the ratio of the area of the triangle \(DEF\) to the area of \(ABC\) is \(R:2r\).

Sum to \(n\) terms the series

1. [(i)] \(\tan\alpha + 2\tan 2\alpha + 2^2\tan 2^2\alpha + \dots\).
2. [(ii)] \(\sin\alpha + 2\sin 2\alpha + 3\sin 3\alpha + \dots\).

Prove that, if \(\cos 2\theta + i\sin 2\theta = p\) and \(\cos 2\phi + i\sin 2\phi=q\), then \[ 2\cos(\theta-\phi) = \sqrt{\frac{p}{q}} + \sqrt{\frac{q}{p}} \text{ and } 2\sin(\theta-\phi) = \frac{1}{i}\left(\sqrt{\frac{p}{q}} - \sqrt{\frac{q}{p}}\right). \] Prove that \(\log \frac{\cos(x+iy)}{\cos(x-iy)} = 2i \tan^{-1}(\tan x \tanh y)\).