Find an expression for \(\tan n\theta\) in terms of \(\tan\theta\), where \(n\) is a positive integer. Prove that \[ \tan^2 20^\circ + \tan^2 40^\circ + \tan^2 80^\circ = 33, \] and that \[ \tan 20^\circ . \tan 40^\circ . \tan 80^\circ = \sqrt{3}. \]
Express \(\frac{2x^3+x^2+2}{(x^2-1)(x^2+2x+2)}\) as the sum of three partial fractions.
Determine the range of values for which the infinite series \[ x - \frac{x^2}{2} + \frac{x^3}{3} - \dots + (-1)^{n-1}\frac{x^n}{n} \dots \] is convergent. Prove that in the approximate formula used by Napier for calculating logarithms \[ \log_e \frac{a}{b} = \frac{a-b}{2}\left(\frac{1}{a}+\frac{1}{b}\right), \] where \(\frac{a-b}{a}\) is small, the error is of the order \(\frac{1}{6}\left(\frac{a-b}{a}\right)^3\).
If the coefficients of the series \(u_0+u_1x+u_2x^2+\dots\) are connected by the relation \(u_{n+2}+au_{n+1}+bu_n=0\), where \(a,b\) are independent of \(n\), find \(u_n\), when \(u_0\) and \(u_1\) are given. Assuming that the series \(1+8x+14x^2+62x^3+\dots\) is of this form, find the coefficient of \(x^n\).
While ascending a tower it is found that at a height \(a\) from the ground the breadth of a river subtends an angle \(\alpha\), and that at a height \(b\) it also subtends the same angle. Prove that the width of the river is \((a+b)\tan\alpha\) and the distance of the tower from the middle of the river is \(\left\{ab+\frac{1}{4}(a+b)^2\tan^2\alpha\right\}^{\frac{1}{2}}\).
Express the area of a triangle in terms of the angles and the radius of the inscribed circle. Prove that the angle at which the perpendicular from the vertex \(A\) to the side \(BC\) of a triangle \(ABC\) cuts the inscribed circle is equal to \[ \cos^{-1}\{\sin\tfrac{1}{2}(B-C)\csc\tfrac{1}{2}A\}. \]
Find the sum of \(n\) terms of the series \[ \sin\theta+\sin(\theta+\alpha)+\sin(\theta+2\alpha)+\dots \] Show that \[ \sin\theta+2\sin 2\theta+\dots+n\sin n\theta = \frac{(n+1)\sin n\theta-n\sin(n+1)\theta}{2(1-\cos\theta)}. \]
Show that the equation of the tangent at any point of the curve \(x=a(\theta+\sin\theta\cos\theta)\), \(y=a\cos^2\theta\) is \((x-a\theta)\tan\theta+y=a\). Prove that the area of the triangle whose sides are this tangent and the coordinate axes is a minimum when \(\theta=\cos 2\theta\cot\theta\).
Obtain the coordinates of the centre of curvature for any point of the curve \(y=f(x)\). Find the ordinates of the real points of the curve \[ y=b\cos(x/a) \] for which the centre of curvature lies on the axis of \(x\).
Trace the curve \(y^2(a+x)=a^2(a-x)\), and show that the volume obtained by rotating it round the line \(x+a=0\) is \(2\pi^2a^3\).