If \[ \frac{\cos(\alpha-3\theta)}{\cos^3\theta} = \frac{\sin(\alpha-3\theta)}{\sin^3\theta} = m, \] prove that \[ m^2+m\cos\alpha=2. \]
A gun is fired from a fort \(A\), and the intervals between seeing the flash and hearing the report at two stations \(B,C\) are \(t,t'\) respectively. \(D\) is a point in \(BC\) produced at a known distance \(a\) from \(A\). Prove that if \(BD=b\) and \(CD=c\), the speed of sound is \[ \left\{\frac{(b-c)(a^2-bc)}{bt'^2-ct^2}\right\}^{\frac{1}{2}}. \] Examine the case when \(a^2=bc\).
In a triangle \(ABC\), with the usual notation, prove that \[ r=4R\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}. \] A circle is drawn touching the circumscribed circle of the triangle internally, and also touching \(AB, AC\). Prove that its radius is \[ r\sec^2\frac{A}{2}. \]
If \[ \tan\alpha = \cos2\omega\cdot\tan\lambda, \] prove that \[ \lambda-\alpha = \tan^2\omega\cdot\sin2\alpha + \frac{1}{2}\tan^4\omega\cdot\sin4\alpha + \frac{1}{3}\tan^6\omega\cdot\sin6\alpha + \dots \]
Express \(\tan n\theta\) in terms of \(\tan\theta\). Prove that the values of \(x\) which satisfy the equation \[ 1-nx - \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \dots + (-1)^{\frac{1}{2}n(n-1)}x^n=0 \] are given by \[ x=\tan\frac{(4r+1)\pi}{4n}, \] where \(r\) is any integer.
Assuming that \(x\{\log(1+x)\}^{-1}\) can be expanded in ascending powers of \(x\), find the first four terms in the expansion. Hence show that a capital sum accumulating at compound interest at \(r\) per cent. per annum will be increased tenfold after \(\left(\frac{230.26}{r}+1.15\right)\) years.
Sum the series:
Eliminate \(\alpha, \beta, \gamma\) from the equations: \begin{align*} \cos\alpha+\cos\beta+\cos\gamma &= l, \\ \sin\alpha+\sin\beta+\sin\gamma &= m, \\ \cos2\alpha+\cos2\beta+\cos2\gamma &= p, \\ \sin2\alpha+\sin2\beta+\sin2\gamma &= q. \end{align*}
Prove that if \(n\) is a positive integer, \[ \cos nx - \cos n\theta = 2^{n-1}\prod_{r=0}^{n-1}\left\{\cos x - \cos\left(\theta+\frac{2r\pi}{n}\right)\right\}. \] Deduce a product for \(\sin n\theta\). Also show that \[ \cos\frac{\pi}{n}\cos\frac{2\pi}{n}\dots\cos\frac{(2n-1)\pi}{n} = \frac{(-1)^n-1}{2^{2n-1}}. \]
Investigate the maxima and minima of the function \((x+1)^5/(x^5+1)\) and trace its graph.
Prove that the equation \((x+1)^5=m(x^5+1)\) has three real roots if \(0