Express \(x^{2n}-2x^n\cos n\theta+1\) as the product of \(n\) real quadratic factors and deduce that \[ \cos n\theta = 2^{n-1} \sin\left(\theta+\frac{\pi}{2n}\right) \sin\left(\theta+\frac{3\pi}{2n}\right)\dots\sin\left(\theta+\frac{2n-1}{2n}\pi\right). \] Prove that \[ \cos\frac{2\pi}{7}+\cos\frac{4\pi}{7}+\cos\frac{6\pi}{7} = -\frac{1}{4}. \]
Expand \(\cos x\) in ascending powers of \(x\), and prove that \[ \cos x \cosh x = 1 - \frac{2^2x^4}{4!} + \frac{2^4x^8}{8!} - \dots. \]
Find the equations of the bisectors of the angles between the straight lines \[ ax+by=c \quad \text{and} \quad bx+ay=d. \] Find the coordinates of the centre of the inscribed circle of the triangle the equations of whose sides are \[ x+y=1, \quad x-y=3 \quad \text{and} \quad 17x+7y+3=0. \]
The straight line \(\frac{x-h}{\cos\alpha} = \frac{y-k}{\sin\alpha}\) through the point \(P\), whose coordinates are \((h,k)\), meets the parabola \(y^2=4ax\) in the points \(Q_1, Q_2\). Obtain a quadratic equation whose roots are the lengths of \(PQ_1, PQ_2\). If \(P\) is on the parabola prove that the length of \(Q_1Q_2\) is \[ \frac{4a\sin(\psi-\alpha)}{\sin^2\alpha\sin\psi}, \] where \(\psi\) is the inclination of the tangent at \(P\) to the axis of \(x\).
Find the coordinates of the point of intersection of the normals to the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) at two points whose eccentric angles are \(\phi_1\) and \(\phi_2\). Prove that the normals at the two points in which the ellipse is cut by the straight line \[ \frac{x\cos\phi}{a^3} + \frac{y\sin\phi}{b^3} + \frac{1}{a^2-b^2}=0 \] meet on the ellipse at the point whose eccentric angle is \(\phi\).
Prove that the conic, whose equation in areal coordinates is \[ lx^2+my^2+nz^2+2pyz+2qzx+2rxy=0, \] is a rectangular hyperbola if \[ (m+n-2p)\cot A + (n+l-2q)\cot B + (l+m-2r)\cot C = 0. \] Shew that \((x-y-z)^2+4xz(1+\tan B\cot C)=0\) is a rectangular hyperbola and find the coordinates of its centre.
If \(y = \sqrt{1-x^2}.\sin^{-1}x\), prove that
Find the equations of the tangent and normal at the point \((h,k)\) of the curve whose equation is \(4x^3=27ay^2\). Shew that coordinates of the centre of curvature at the point \((h,k)\) are \[ \left(-h-\frac{h^2}{2a}, 4k+\frac{9ak}{h}\right). \]
Evaluate \[ \int (1+x)\sqrt{1-x^2}dx, \quad \int_0^\pi \cos 2\theta \log(1+\tan\theta)d\theta, \quad \int_0^{\pi/2} \sin^2\theta\cos 2\theta d\theta. \] Prove that \[ (m+np)\int x^{m-1}(x^n+a^n)^p dx = x^m(x^n+a^n)^p + npa^n\int x^{m-1}(x^n+a^n)^{p-1}dx. \]
Prove that the area of the loop of the curve \(y^2(a+x)=x^2(a-x)\) is \(2a^2(1-\frac{\pi}{4})\) and that the volume formed by the revolution of the loop about the axis of \(x\) is \(2\pi a^3(\log 2 - \frac{2}{3})\).