Given the sides of a triangle, find an expression for the tangent of half of one of its angles. With the usual notation prove that, if \(\tan\frac{1}{2}A+\tan\frac{1}{2}B+\tan\frac{1}{2}C=2\), \[ 4R+r=a+b+c. \]
\(ABCD\) is a parallelogram. \(P, Q, R, S\) are four points taken respectively on the sides \(CD, CB, AB, AD\) produced, such that \(PAQ\) and \(RCS\) are straight lines. Prove that \(PS\) and \(QR\) are parallel.
Shew that the length of a chord of a parabola drawn through the focus \(S\) parallel to the tangent at \(P\) is \(4SP\). Prove that the normal at any point, terminated at the axis, is a mean proportional between the segments of the focal chord to which it is perpendicular.
Find the condition that the general equation of the second degree should represent two straight lines. Prove that if the equation \(3x^2+2hxy+2y^2+5x+5y+2=0\) represents two straight lines, the tangent of the angle between them is either 1 or \(\frac{1}{7}\). Find the equations of the lines in each case.
Define a differential coefficient, and shew that if \(\frac{dy}{dx}\) is positive for any value of \(x\), the value of \(y\) is increasing as \(x\) increases through that value.
Prove that
\[ \tan x > x + \frac{x^3}{3}. \quad \left(0
Find the \(n\)th differential coefficient of \(x\log(1+x)\).
State Maclaurin's Theorem on the expansion of \(f(x)\) in a series of ascending powers of \(x\). Prove that if \(\{\log(1+x)\}^2 = a_2x^2 - a_3x^3 + a_4x^4 - a_5x^5 \dots\) \[ n\{(n+1)a_{n+1}-na_n\} = 2, \] and find \(a_n\).
Integrate with respect to \(x\) \[ \frac{x^2+1}{x+2}, \quad \frac{(a^x+b^x)^2}{a^x b^x}, \quad \cos^2 2x \sin 3x. \]
Prove that the sum of the squares on the four sides of a quadrilateral is equal to the sum of the squares on the two diagonals together with four times the square on the line joining the middle points of the diagonals.
Prove that the two tangents from a point to a conic subtend equal or supplementary angles at the focus. \(O\) is any point on a chord \(PQ\) of a conic the tangents at whose extremities meet in \(T\). \(ON\) is perpendicular to the directrix, and the perpendicular from \(O\) to \(ST\) (\(S\) being the focus) meets \(SQ\) in \(L\). Prove that \(SL = e \cdot ON\), where \(e\) is the eccentricity.