Find formulae of reduction for \[ (1) \int \frac{dx}{(a^2+x^2)^n}, \quad (2) \int \frac{dx}{(a+b\tan x)^n}. \]
Prove that, in the curve \(y^2(a+x)=x^2(a-x)\), the area between the curve and its asymptote and the area of the loop are in the ratio \(4+\pi:4-\pi\).
A circle \(C\) has its centre on the circumference of another circle \(C'\). Any tangent to \(C\) cuts \(C'\) in two points \(Q, R\) and the other tangents to \(C\) from \(Q\) and \(R\) are drawn to cut \(C'\) in \(Q', R'\) respectively. Prove that \(Q'R'\) touches \(C\).
Two tetrahedra are such that lines joining corresponding vertices meet in a point, prove that pairs of corresponding edges meet in points all lying in one plane.
\(O\) is the centre of a rectangular hyperbola and \(P, Q\) are two points on it. The tangents at \(P, Q\) intersect \(OQ, OP\) in \(P'\) and \(Q'\) respectively and intersect each other in \(T\). Prove that \(O, P', T, Q'\) lie on a circle.
A conic passes through three given points. If one asymptote is in a fixed direction, prove that the other touches a fixed parabola which touches the three sides of the triangle formed by the given points.
If \(a+b+c=0\) and \(x+y+z=0\), prove that \[ a^2x^2+b^2y^2+c^2z^2-bcyz-cazx-abxy = \frac{1}{4}(a^2+b^2+c^2)(x^2+y^2+z^2). \]
Prove that the product of the infinite periodic continued fractions \[ \frac{1}{a_1+} \frac{1}{a_2+} \frac{1}{a_3+\dots} \frac{1}{a_n+} \frac{1}{a_1+} \frac{1}{a_2+\dots} \] and \[ \frac{1}{a_n+} \frac{1}{a_{n-1}+} \frac{1}{a_{n-2}+\dots} \frac{1}{a_1+} \frac{1}{a_n+} \frac{1}{a_{n-1}+\dots} \] is \(p_n/q_{n-1}\) where \(p_r/q_r\) is the \(r\)th convergent of \[ \frac{1}{a_1+} \frac{1}{a_2+\dots} \frac{1}{a_n}. \]
Having given \[ \sin\phi = k\tan\frac{\theta+\psi}{2} \text{ and } \sin\psi = k\tan\frac{\theta+\phi}{2}, \] prove that \[ \sin\theta = k\tan\frac{\phi+\psi}{2}. \]
\(I\) is the centre of the inscribed circle of a triangle \(ABC\) and \(D, E, F\) are the feet of the perpendiculars from \(I\) on the sides \(BC, CA, AB\) respectively. The radii of the circles inscribed in the quadrilaterals \(AEIF, BFID\) and \(CDIE\) are \(\rho_1, \rho_2, \rho_3\) respectively and \(r\) is the radius of the circle inscribed in the triangle. Prove that \[ (r-2\rho_1)(r-2\rho_2)(r-2\rho_3) = r^3-4r\rho_1\rho_2\rho_3. \]