Prove that, when \(n\) is an odd integer, \[ \frac{\sin n\theta}{n\sin\theta} = 1 - \frac{n^2-1^2}{3!}\sin^2\theta + \frac{(n^2-1^2)(n^2-3^2)}{5!}\sin^4\theta - \text{etc.} \] and shew that \begin{align*} \sin^2\frac{\pi}{11}+\sin^2\frac{2\pi}{11}+\sin^2\frac{3\pi}{11}+\sin^2\frac{4\pi}{11}+\sin^2\frac{5\pi}{11} &= \frac{11}{4}; \\ \cot^2\frac{\pi}{11}+\cot^2\frac{2\pi}{11}+\cot^2\frac{3\pi}{11}+\cot^2\frac{4\pi}{11}+\cot^2\frac{5\pi}{11} &= 15. \end{align*}
State carefully Demoivre's Theorem. Find all the cube roots of \(88+16\sqrt{-1}\), having given that, when \(\tan\theta=2\), \(\tan 3\theta=\frac{2}{11}\).
Prove that, if a circle cuts two circles orthogonally, its centre lies on their radical axis. Prove also that if two circles \(S\) and \(S'\) are orthogonal, every diameter of \(S\) cuts \(S'\) in inverse points with respect to \(S\).
\(S\) is a focus of a conic, and the tangent at \(P\) meets the corresponding directrix in \(R\). Prove that the angle \(PSR\) is a right angle. Find the locus of the focus of a variable conic with a given line as the corresponding directrix, and touching another given line at a given point. Indicate the parts of the locus which correspond respectively to a parabola, an ellipse or a hyperbola.
\(C\) is the centre and \(ACA'\) the major axis of an ellipse. The tangent at \(P\) meets \(CA\) produced in \(T\). \(PN\) is perpendicular to \(CA\). Prove that \[ CN \cdot CT = CA^2. \] \(CY\) and \(AZ\) are drawn perpendicular to \(PT\). Prove that \[ CA \cdot AZ = CY \cdot AN. \]
Explain very briefly the principles of orthogonal projection. \(ABCD\) is a rhombus of side 2 inches; the angle \(A\) is \(60^\circ\). The rhombus is the orthogonal projection of a square in another plane. Prove that the side of the square is \(\sqrt{6}\) inches, and find the angle between the planes.
Prove that the shortest distance between two non-intersecting straight lines is perpendicular to both of them. Prove that the lines joining the middle points of opposite edges of a regular tetrahedron intersect at right angles.
Find the angle between the straight lines whose equation is \[ ax^2+2hxy+by^2=0. \] Prove that the equation \[ (a+2h+b)x^2 + 2(a-b)xy+(a-2h+b)y^2=0 \] denotes a pair of straight lines each inclined at \(45^\circ\) to one or other of the lines given by \[ ax^2+2hxy+by^2=0. \]
Find the equation of the polar of the point \((h, k)\) with respect to the circle \[ x^2+y^2+2gx+2fy+c=0. \] Prove that the locus of the pole of the line \(y=k\) with respect to the family of coaxal circles \[ x^2+y^2+2\lambda x + c = 0, \] in which \(c\) is constant but \(\lambda\) varies, is the parabola \[ x^2=ky+c. \]
Find the condition that \[ y=mx+c \] should be a normal to the parabola \[ y^2=4ax. \] Pairs of tangents are drawn to the parabola from points on a given fixed straight line. Prove that the locus of the intersection of the normals at the points of contact is a parabola.