If \(R\) and \(r\) are the radii of the circumscribed and inscribed circles of a triangle \(ABC\), prove that \[ r=4R\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}. \] Show also that the radius of the circle inscribed in the triangle formed by joining the centres of the three escribed circles is \[ \frac{4R\cos\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2}}{\cos\frac{A}{2}+\cos\frac{B}{2}+\cos\frac{C}{2}}. \]
Find the sum of \(n\) terms of the series:
Express \(x^{2n}-2x^n\cos n\theta+1\) as the product of \(n\) real quadratic factors, and deduce that \[ \sin n\theta = 2^{n-1}\sin\theta\sin\left(\theta+\frac{\pi}{n}\right)\sin\left(\theta+\frac{2\pi}{n}\right)\dots\sin\left(\theta+\frac{n-1}{n}\pi\right). \] Prove that \[ \sin\frac{\pi}{7}\sin\frac{2\pi}{7}\sin\frac{3\pi}{7} = \frac{\sqrt{7}}{8}. \]
Prove that the equation of the straight lines, which bisect the angles between the straight lines whose equation is \(ax^2+2hxy+by^2=0\), is \[ \frac{x^2-y^2}{a-b}=\frac{xy}{h}. \] Prove that one of the three straight lines given by the equation \[ x^3+3(1-k^2)xy^2+k(k^2-3)y^3=0 \] bisects the angle between the other two.
Shew that, if \(x=at^2+bt\) and \(y=ct+d\), where \(t\) is a variable parameter, the locus of the point \((x,y)\) is a parabola whose latus rectum is \(c^2/a\); and find the coordinates of its focus.
Find the condition that the straight line \(\dfrac{x-h}{\cos\theta} = \dfrac{y-k}{\sin\theta}\) is a tangent to the ellipse \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1. \] Hence find the equation of the two tangents to the ellipse, which can be drawn from \((h,k)\), in the form \[ \left( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} \right) \left( \frac{h^2}{a^2} + \frac{k^2}{b^2} - 1 \right) = \left( \frac{h(x-h)}{a^2} + \frac{k(y-k)}{b^2} \right)^2. \]
Find the condition that a conic whose equation, in areal coordinates, is \[ lyz + mzx + nxy = 0 \] should be a rectangular hyperbola. Shew that the conic \((b^2-c^2)yz + b^2zx - c^2xy=0\) is a rectangular hyperbola whose centre is the middle point of the side \(BC\) of the triangle of reference \(ABC\).
Find the maximum and minimum values of the expression \(\dfrac{2x^2-7x+3}{x-5}\). Shew that the least value of the sum of the squares of the perpendiculars from a point within a triangle \(ABC\) on the three sides is \(\frac{1}{2}(a^2+b^2+c^2) - \dfrac{a^4+b^4+c^4}{2(a^2+b^2+c^2)}\).
Prove that the distance from the origin of the centre of curvature at any point of a curve is \(\left[ \left(\frac{dp}{d\psi}\right)^2 + \left(\frac{d^2p}{d\psi^2}\right)^2 \right]^{\frac{1}{2}}\), where \(\psi\) is the inclination of the tangent at the point to the initial line and \(p\) the perpendicular from the origin on the tangent. Find the value of \(p\) in terms of \(\psi\) for the cardioid \(r=a(1-\cos\theta)\) and determine the distance from the origin of the centre of curvature at the point \(\theta=\dfrac{\pi}{2}\).
Evaluate \(\displaystyle\int \sec^3 x dx\), \(\displaystyle\int x^2 \sin^2 x dx\), \(\displaystyle\int \frac{(x-1)(x-4)}{(x-2)(x-3)} dx\). Prove that \[ \int_0^{\frac{\pi}{2}} \cos^3 x \sin 5x = \frac{1}{2}. \]