Problems

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 \(\log \frac{\cos(x+iy)}{\cos(x-iy)} = 2i \tan^{-1}(\tan x \tanh y)\).

Find the length of the perpendicular from the points \((h,k)\) on the straight line \(x\cos\alpha+y\sin\alpha=p\). Shew that \(x^2(k^2-p^2)+y^2(h^2-p^2) = 2hkxy\) is the equation of two straight lines through the origin such that the lengths of the perpendiculars on them from \((h,k)\) are each equal to \(p\).

Shew that, by a proper choice of axes, the equations of any two circles may be written in the form \(x^2+y^2+2ax+b=0\) and \(x^2+y^2+2a'x+b=0\). If the polars of a point \(P\) with respect to these two circles meet in \(Q\), shew that the axis of \(y\) bisects \(PQ\).

Find the equations of the tangent and normal to the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) at the point whose eccentric angle is \(\phi\). Prove that the envelope of the line joining the points of contact of two perpendicular tangents to an ellipse is another ellipse.