Prove that \[ 1+\sec 20^\circ = \cot 30^\circ \cot 40^\circ \] and solve the equation \[ 1+\sec\theta = \tan 3\theta \cot 2\theta. \]
On opposite sides of a base \(BC\) are described two triangles \(ABC, BCD\), such that \(\angle ABC=30^\circ, \angle ACB=80^\circ, \angle DBC = \angle DCB = 50^\circ\). Shew that, if \(AD\) is drawn, \(\angle DAC=30^\circ\), and find the other angles of the figure.
When are two ranges said to be homographic? Shew that two homographic ranges on the same straight line have two common points, real, coincident, or imaginary. \par In what case is one of the common points at infinity?
Prove that the tangents from a point \(O\) to a conic subtend angles at a focus which are equal or supplementary. \par Shew that the same proposition is true if instead of a focus is taken the foot of the perpendicular from \(O\) on the major axis.
Prove that two of the tangents of the parabola \(y^2=ax\) are identical with two of the tangents of \(a^2x^2=c^2(4y-3c)\), and find their equations. Prove also that the two curves touch one another.
Prove that the two circles \[ (x-\alpha)^2+(y-\beta)^2 = \lambda(x^2+y^2), \quad (\alpha+\mu\beta)(x^2+y^2) = (\alpha^2+\beta^2)(x+\mu y) \] cut orthogonally, and interpret the result geometrically.
Find the equation of the tangent at any point of a given curve. \par Prove that in the lemniscate, \((x^2+y^2)^2=2a^2(x^2-y^2)\), the tangent at a point for which \(y=\frac{2}{3}x\) is \(117y-44x=7a\sqrt{14}\).
Integrate the following expressions with respect to \(x\) \[ \frac{1}{\sqrt{(x^2-a^2)}}, \quad \frac{1}{2\sqrt{\{(2-x)(x-1)\}}}, \quad \frac{20}{25+9\cos x+12\sin x}. \]
Given two circles and a point \(A\) on one of them, shew how to draw a chord \(BA\) of one circle such that if produced to meet the other circle in \(C\), \(BA\) may be equal to \(AC\).
Prove that the inverse of a circle is in general another circle. \par If \(P, Q\) are inverse points with regard to a circle \(A\) and the figure is inverted with regard to any centre, \(P', Q'\) and \(A'\) being the corresponding elements in the inverse figure, then \(P', Q'\) are inverse points with regard to \(A'\).