S L Loney Solutions for Chapter: Relations Between the Sides and the Trigonometrical Ratios of the Angles of a Triangle, Exercise 2: Examples XXVII

The sides of a triangle are in $A.P.$ and the greatest and the least angles are $\mathit{\text{θ}}$ and $\varphi$, prove that $4\left(1-\cos \theta \right)\left(1-\cos \varphi \right)=\cos \theta +\cos \varphi$.

 The sides of a triangle are in $\mathrm{A}.\mathrm{P}.$ and the greatest angle exceeds the least by $90°;$ prove that the sides are proportional to $\sqrt{7}+1, \sqrt{7}$ and $\sqrt{7}-1.$

If in a $△ABC, \angle C=60°,$ then prove that $\frac{1}{a+c}+\frac{{\displaystyle 1}}{{\displaystyle b+c}}=\frac{{\displaystyle 3}}{{\displaystyle a+b+c}}$.

In any triangle $ABC$, if $D$ be any point of the base $BC$, such that $BD\mathbf{:}DC\mathbf{:}\mathbf{:}m\mathbf{:}n$ and if $\mathrm{\angle }BAD=\alpha , \mathrm{\angle }\mathrm{DAC}=\beta , \mathrm{\angle }CDA=\theta$ and $AD=x$, prove that $\left(m+n\right)\cot \theta =m\cot \alpha -n\cot \beta =n\cot B-m\cot C$ and ${\left(m+n\right)}^2\mathbf{:}{x}^2=\left(m+n\right)\left(m{b}^2+n{c}^2\right)-mn{a}^2$.

If in a triangle the bisector of the side $\mathrm{c}$ be perpendicular to the side $\mathrm{b}$, prove that $2\tan A+\tan C=0$.

In any triangle prove that if $\theta$ be any angle, then $\mathrm{bcos}\theta =c\cos \left(A-\theta \right)+a\cos \left(C+\theta \right)$.

If $p$ and $q$ be the perpendiculars from the angular points $A$ and $B$ on any line passing through the vertex $C$ of the triangle $ABC$, then prove that
$a^2{p}^2+b^2{q}^2-2abpq\cos C=a^2{b}^2{\sin }^{2}C$.

In the triangle $ABC,$ lines $OA,$ $OB$ and $OC$ are drawn so that the angle $OAB,$ $OBC$ and $OCA$ are each equal to $\omega ;$ prove that $\cot \omega =\cot A+\cot B+\cot C$ and ${\mathrm{cosec}}^2\text{ω}={\mathrm{cosec}}^{2}A+{\mathrm{cosec}}^{2}B+{\mathrm{cosec}}^{2}C.$