Prove that \[ \sin A+\sin B+\sin C - \sin(A+B+C) = 4\sin\tfrac{1}{2}(B+C)\sin\tfrac{1}{2}(C+A)\sin\tfrac{1}{2}(A+B). \] Shew that, if \[ \sin 2A+\sin 2B+\sin 2C = 4\sin A\sin B\sin C + 2\sin(A+B+C), \] either \(A+B+C\) is an odd multiple of \(\pi\), or one of the quantities \((B+C-A), (C+A-B), (A+B-C)\) is an even multiple of \(\pi\).
If \(\theta_1\) and \(\theta_2\) are two values of \(\theta\), not differing by a multiple of \(\pi\), which satisfy the equation \[ a\cos(\theta+\alpha)+b\sin(\theta-\alpha)=c, \] prove that \[ \cos(\theta_1+\theta_2) = \frac{(a^2-b^2)\cos 2\alpha}{a^2+b^2-2ab\sin 2\alpha} \] and \[ \sec^2\frac{\theta_1-\theta_2}{2} = \frac{1}{c^2}(a^2+b^2-2ab\sin 2\alpha). \]
Prove that the area of a triangle \(ABC\) is \(2R^2\sin A\sin B\sin C\), where \(R\) is the radius of the circumscribed circle; and, if \(\Delta\) be the area of the triangle \(ABC\), the perimeter of the pedal triangle is \(2\Delta/R\).
Prove that \[ \sin n\theta = 2^{n-1}\sin\theta\sin\left(\theta+\frac{\pi}{n}\right)\sin\left(\theta+\frac{2\pi}{n}\right)\dots\sin\left(\theta+\frac{(n-1)\pi}{n}\right). \] Shew that the coefficient of \(x^n\) in the expansion of \((1-2x\cos\alpha+x^2)^{-1}\) is \[ \frac{\sin(n+1)\alpha}{\sin\alpha}. \]
Find the tangent of the angle between the two straight lines whose equation is \[ ax^2+2hxy+by^2=0. \] If these two straight lines are the sides of a parallelogram of which one diagonal is the straight line \(lx+my=1\), shew that the other diagonal is the straight line \[ x(am-hl)+y(hm-bl)=0. \]
Find the condition that the straight line \(x\cos\alpha+y\sin\alpha=p\) should be a tangent to the parabola \(y^2=4ax\), and find the equation of the normal at the point of contact. The normal at \(P\) meets the axis at \(G\) and the parabola again at \(P'\), and the normal at \(P'\) meets the axis at \(G'\); \(GQ\) is drawn perpendicular to the axis and equal to \(GG'\). Prove the locus of \(Q\) is the curve \[ xy-4ax-2ay+4a^2=0. \]
Find the equation of the straight line joining two points on the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\), whose eccentric angles are \(\alpha\) and \(\beta\). Deduce the equation of the tangent and normal at a point whose eccentric angle is given, and find the coordinates of the point of intersection of the normals at the points whose eccentric angles are \(\alpha\) and \(\beta\). Hence find the coordinates of the centre of curvature at either point.
Explain the meaning of the equation \(\alpha\beta=\gamma^2\), where \(\alpha=0, \beta=0, \gamma=0\) are the equations of straight lines. Shew that the straight lines \(\alpha=\mu\gamma\) and \(\mu\beta=\gamma\) meet on the curve \(\alpha\beta=\gamma^2\), and find the equation of the tangent to the curve at this point.
Shew that the conic, whose equation in areal coordinates is \[ \sqrt{lx}+\sqrt{my}+\sqrt{nz}=0, \] touches the sides of the triangle of reference, and that the coordinates of its centre are proportional to \(m+n, n+l, l+m\). Hence deduce the equation of the circle inscribed in the triangle of reference.
State and prove Leibnitz's Theorem for finding the \(n\)th differential coefficient of the product of two functions of the variable. Prove that, if \(y=e^{\tan^{-1}x}\), \[ (1+x^2)\frac{d^{n+1}y}{dx^{n+1}} = (1-2nx)\frac{d^ny}{dx^n}-n(n-1)\frac{d^{n-1}y}{dx^{n-1}}. \]