Problems

Prove that

1. [(i)] \(\cos36^\circ-\cos72^\circ=\frac{1}{2}\).
2. [(ii)] \(\tan^{-1}\frac{1}{3}+\tan^{-1}\frac{1}{5}+\tan^{-1}\frac{1}{7}+\tan^{-1}\frac{1}{8}=\frac{\pi}{4}\).

Express \[ 1-\cos^2x-\cos^2y-\cos^2z+2\cos x\cos y\cos z \] as the product of four sines.

If \(a,b,c,d\) are the sides (taken in order) of a quadrilateral inscribed in a circle, prove that the area of the quadrilateral is equal to \(\sqrt{(s-a)(s-b)(s-c)(s-d)}\), where \(2s=a+b+c+d\). Prove also that the diagonals are in the ratio of \(ab+cd\) to \(ad+bc\).

Sum the following series:

1. [(i)] \(\sin a+\sin3a+\sin5a+\dots\) to \(n\) terms.
2. [(ii)] \(\sin\theta+2\sin2\theta+3\sin3\theta+\dots\) to \(n\) terms.
3. [(iii)] \(x\sin a - \frac{1}{3}x^3\sin3a+\frac{1}{5}x^5\sin5a-\dots\) to infinity. (\(x<1\))

Prove that \(\displaystyle\frac{\sin n\theta}{\sin\theta}\) is divisible by \(\cos\theta-\cos\alpha\), where \(n\) is an integer, and shew that \[ \frac{\sin n\theta}{\sin\theta} = 2^{n-1}\left(\cos\theta-\cos\frac{\pi}{n}\right)\left(\cos\theta-\cos\frac{2\pi}{n}\right)\dots\left(\cos\theta-\cos\frac{n-1}{n}\pi\right). \] Prove that \[ 32\cos\frac{\pi}{11}\cos\frac{2\pi}{11}\cos\frac{3\pi}{11}\cos\frac{4\pi}{11}\cos\frac{5\pi}{11}=1. \]

The diagonals of a parallelogram are the straight lines whose equation referred to rectangular coordinates is \[ ax^2+2hxy+by^2=0, \] and \((\alpha,\beta)\) are the coordinates of the middle point of one side of the parallelogram. Find the equation of that side and shew that one of the angles of the parallelogram is \[ \tan^{-1}\frac{a\alpha^2+2h\alpha\beta+b\beta^2}{h(\alpha^2-\beta^2)+\alpha\beta(b-a)}. \]

Find the equations of the tangent and normal at the point \((h,k)\) on the ellipse \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1. \] If the tangent at \(P\) cuts the directrices in \(Q\) and \(Q'\), prove that the other tangents from \(Q\) and \(Q'\) to the ellipse meet on the normal at \(P\).

Find the coordinates of the focus of the parabola \[ (x\sin\theta+y\cos\theta)^2=4ay\sin\theta, \] referred to rectangular axes; and shew that the equation of the latus rectum is \[ x\cos\theta-y\sin\theta=a\cos2\theta. \]

Find the conditions that the equation \[ Ax^2+By^2+Cz^2+2Fyz+2Gzx+2Hxy=0, \] in areal coordinates, represents a circle. Prove that, if \(ABC\) is the triangle of reference, \[ (y\cot B-z\cot C)^2=x(y+z) \] is the equation of the circle described on the perpendicular from \(A\) on the side \(BC\), as diameter; and find the coordinates of the point of intersection of the tangents to this circle at the points at which it cuts the sides \(AB\) and \(AC\).

Prove that, if \[ y=(\sin^{-1}x)^2, \] then \[ (1-x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} = 2, \] and \[ (1-x^2)\frac{d^{n+2}y}{dx^{n+2}} - (2n+1)x\frac{d^{n+1}y}{dx^{n+1}} - n^2\frac{d^ny}{dx^n}=0. \]

The two chains of a suspension bridge hang in a parabola of span 80' and dip 16'; they are stiffened by a roadway, rigid except for hinges at the centre and at each end. If the chains and roadway weigh 0.3 ton per foot of span, find the maximum tension in each chain when a 2 ton concentrated load crosses the bridge, and the maximum bending moment on the roadway.