Integrate
Trace the curve \(x^2(x^2-a^2)+y^2(x^2+a^2)=0\), and find the area of a loop.
Prove that \[ \tan 20^\circ \tan 30^\circ \tan 40^\circ = \tan 10^\circ. \] If \(A+B+C=90^\circ\), prove that \[ \frac{\tan A}{\tan C} = \frac{1-\cos 2A+\cos 2B+\cos 2C}{1+\cos 2A+\cos 2B-\cos 2C}. \]
The lines joining the angular points of a triangle \(ABC\) to the middle points of the opposite sides intersect in a point \(O\); prove that \[ OA^2+OB^2+OC^2 = \frac{8}{9}R^2(1+\cos A \cos B \cos C), \] where \(R\) is the radius of the circumcircle.
Prove that the sum of \(n-1\) terms of the series \[ \tan\theta\tan 2\theta + \tan 2\theta\tan 3\theta + \tan 3\theta\tan 4\theta + \dots \] is equal to \(\tan n\theta\cot\theta-n\). Sum the series \[ \cos\alpha + \frac{1}{3!}\cos(\alpha+2\beta) + \frac{1}{5!}\cos(\alpha+4\beta) + \dots \text{ to infinity}. \]
Resolve \(x^{2n}-2x^ny^n\cos n\theta+y^{2n}\) into factors. Prove that \[ \sin n\phi = 2^{n-1}\sin\phi \sin\left(\phi+\frac{\pi}{n}\right)\dots\sin\left(\phi+\frac{n-1}{n}\pi\right). \]
Prove that the equation of the straight line joining the feet of the perpendiculars from the point \((h,k)\) on the lines \(ax^2+2bxy+cy^2=0\) is \[ x(ah-ch+2bk)+y(ck-ak+2bh)+ak^2-2bhk+ch^2=0. \]
Find the equation of the normal at \(P\) to the parabola \(y^2=4ax\) in the form \[ y = mx - 2am - am^3, \] and, if the normal cuts the parabola again at \(Q\), prove that \(PQ=4a(1+m^2)^{3/2}/m^2\).
Find the equation of the line joining two points on the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\), whose eccentric angles are \(\phi_1\) and \(\phi_2\); hence deduce the equation of the tangent at a point. If \(\phi_1, \phi_2, \phi_3, \phi_4\) are the eccentric angles of the points in which the ellipse is cut by a circle, prove that \(\phi_1+\phi_2+\phi_3+\phi_4=2n\pi\). Shew that the circle of curvature at a point whose ordinate is \(\frac{1}{2}b\) passes through an extremity of the minor axis of the ellipse.
If the equations of three straight lines are expressed in the forms \(\alpha=0, \beta=0, \gamma=0\), interpret the equation \(\alpha\beta=\gamma^2\), and shew that \(\alpha+2\lambda\gamma+\lambda^2\beta=0\) is a tangent to the curve. Shew that the equation of the tangents from the origin to the conic \[ (x-a)(y-a) = (x+y-b)^2 \] is \[ a^2(x-y)^2+4\{ax+(a-b)y\}\{(a-b)x+ay\}=0. \]