Prove that in the expansion of \((1+x)^m + (1-x)^m\), where \(-1< x < 1\), the terms are either all positive or after a stage become and remain negative, and that in the latter case the last positive term exceeds numerically the sum of the infinite series of negative terms. Deduce or otherwise prove that \((1+x)^m + (1-x)^m - 2\) vanishes with or has the same sign as \(m(m-1)x^2\).
Shew that the sum of the \(r\)th powers of the first \(n\) odd integers, when \(r\) is a positive integer, is the coefficient of \(x^r/r!\) in the expansion of \(e^{nx} \frac{\sinh nx}{\sinh x}\) and that, when \(r\) is odd, the sum can be expressed as a polynomial in \(n^2\). Obtain the sum in the case \(r=3\).
The sine of an acute angle is equal to \(\cdot 9998\), accurately; with the aid of the four-figure tables find the angle to the nearest tenth of a minute.
Obtain the expressions for \(\cos n\alpha\) and \(\sin n\alpha\) in terms of \(a\) where \(a = \cos\alpha + i\sin\alpha\). Prove that \[ \sum \sin(4\alpha-\beta-\gamma)\sin(\beta-\gamma) = 4\prod \sin(\beta-\gamma) \times \sum \sin 2\alpha, \] where the summation and product symbols refer to a cyclic interchange of the angles \(\alpha, \beta, \gamma\).
Prove that \[ \sin 7\theta = \sin\theta(c^3+c^2-2c-1), \quad \text{where } c = 2\cos 2\theta. \] According to a rule used by practical draughtsmen, the side of a regular heptagon inscribed in a circle is equal to the height of an equilateral triangle whose side is the radius. Shew that the error in this approximation is about one-fifth per cent.
Write down the series for \(e^{a/x}\) in descending powers of \(x\), and deduce (or prove by induction) that \[ \frac{d^n}{dx^n}(x^n e^{a/x}) = \frac{1}{x} \frac{d^{n-1}}{dx^{n-1}}\left\{x^2 \frac{d}{dx}(e^{a/x})\right\} = (-1)^n \frac{a^n}{x^{n+1}} e^{a/x}. \] If the common value of these functions is denoted by \(y\), prove that \[ x^2 \frac{dy}{dx} + \{(n+1)x+a\}y = 0. \]
A quadratic function of \(x\) takes the values \(y_1, y_2, y_3\) corresponding to three equidistant values of \(x\); prove that, if \(y_1+y_3>2y_2\), the minimum value of the function is \[ y_2 - \frac{(y_3-y_1)^2}{8(y_1+y_3-2y_2)}. \]
The coordinates of points on a curve are given as functions of a parameter \(\theta\), prove that in general the condition for an inflexion is \(x'y''-y'x''=0\); and that the conditions for a cusp are \(x'=0, y'=0\), where accents denote differentiation with respect to \(\theta\). Shew that the coordinates of a point on the curve \(r=a+b\cos\theta\) can be written in the forms \(x=\frac{1}{2}b+a\cos\theta+\frac{1}{2}b\cos 2\theta\), \(y=a\sin\theta+\frac{1}{2}b\sin 2\theta\); and prove that there are two inflexions if \(a\) lies between \(b\) and \(2b\), while there is a cusp if \(a=b\).
Eliminate \(\theta\) from the equations \[ a\sin\theta + b\cos\theta = a\operatorname{cosec}\theta - b\sec\theta = 1. \] Prove that \[ \tan(\alpha-\beta)+\tan(\beta-\gamma)+\tan(\gamma-\delta)+\tan(\delta-\alpha) + \frac{\sin(\alpha-\beta+\gamma-\delta)\sin(\alpha-\gamma)\sin(\beta-\delta)}{\cos(\alpha-\beta)\cos(\beta-\gamma)\cos(\gamma-\delta)\cos(\delta-\alpha)} = 0. \]
Prove that the length of the line joining the orthocentre of a triangle \(ABC\) to the middle point of the side \(BC\) is \[ \frac{a}{2}\left\{\frac{1-4\cos A \cos B \cos C}{\sin^2 A}\right\}^{\frac{1}{2}}. \] Prove that, if the side \(a\) of a triangle \(ABC\) is increased by a small quantity \(x\) while the other two sides remain constant, the radius of the circumscribing circle will be increased by approximately \[ \frac{1}{2}x \operatorname{cosec} A \cot B \cot C. \]