Application of Sine and Cosine Formula (Properties of Triangles)

If the side-lengths $a, b, c$ are in $A.P.,$ prove that $\cot \frac{1}{2} \mathrm{A},\cot \frac{1}{2} \mathrm{B},\cot \frac{1}{2}\mathrm{C}$ are in $A.P.$

In $\Delta \mathrm{ABC}$, If the side-lengths $a, b$ and $c$ are in $A.P.,$ then prove that $\cos \frac{1}{2}( \mathrm{A}-\mathrm{C})=2\sin \frac{1}{2} \mathrm{B}$.

If $\mathrm{sinA}:\mathrm{sinC}=\sin (\mathrm{A}-\mathrm{B}):\sin (\mathrm{B}-\mathrm{C})$, prove that $a^2,b^2,c^2$ are in $A.P.$

In $\Delta \mathrm{ABC}$ if $a^2,b^2,c^2$ be in $A.P,$ prove that $\mathrm{cotA},\mathrm{cotB},\mathrm{cotC}$ are also in $A.P.$

In $\Delta \mathrm{ABC}$, If $\mathrm{cosA}=\mathrm{sinB}-\mathrm{cosC}$, prove that the triangle is right-angled.

In $\Delta \mathrm{ABC}$, If $(\cos A+2\cos C):(\cos A+2\cos B)=\sin B:\sin C$, prove that the triangle is either isosceles or right- angled.

If $a\mathrm{tanA}+b\tan$ $\mathrm{B}=(a+b)\tan \frac{1}{2}( \mathrm{A}+\mathrm{B})$, prove that the triangle is isosceles.

If $\mathrm{cosB}=\frac{\mathrm{sinA}}{2\mathrm{sinC}}$, prove that the triangle is isosceles.

If ${\mathrm{x}}^2+\mathrm{x}+1, 2\mathrm{x}+1$and ${\mathrm{x}}^2-1$, are lengths of sides of a triangle, then prove that the angle of the highest measure is$120°$.

If ${\mathrm{a}}^4+{\mathrm{b}}^4+{\mathrm{c}}^4=2{\mathrm{c}}^2\left({\mathrm{a}}^2+{\mathrm{b}}^2\right)$. Prove that, $\mathrm{m}\angle \mathrm{ACB}=45°$ or $135°$.

In $\Delta \mathrm{ABC}$ If $a=2b$ and $A=3B$, find the measures of the angles of the triangle.

In $\Delta \mathrm{ABC}$, prove that If $\frac{1}{\mathrm{a}+\mathrm{c}}+\frac{1}{\mathrm{b}+\mathrm{c}}=\frac{3}{\mathrm{a}+\mathrm{b}+\mathrm{c}}$, then prove $\mathrm{C}=60°$.

 In $\Delta \mathrm{ABC}$, prove that ${\mathrm{a}}^2\mathrm{cotA}+{\mathrm{b}}^2\mathrm{cotB}+{\mathrm{c}}^2\mathrm{cotC}=4\mathrm{\Delta }$.

In $\Delta \mathrm{ABC}$, prove that $\mathrm{cotA}+\mathrm{cotB}+\mathrm{cotC}=\frac{{\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2}{4\mathrm{\Delta }}$

In $\Delta \mathrm{ABC}$, prove that $(\mathrm{b}-\mathrm{c})\cot \frac{\mathrm{A}}{2}+(\mathrm{c}-\mathrm{a})\cot \frac{\mathrm{B}}{2}+(\mathrm{a}-\mathrm{b})\cot \frac{\mathrm{C}}{2}=0.$

In $\Delta \mathrm{ABC}$, prove that $1-\tan \frac{\mathrm{A}}{2}·\tan \frac{\mathrm{B}}{2}=\frac{\mathrm{c}}{\mathrm{s}}$.

In $\Delta \mathrm{ABC}$ prove that $(a-b{)}^2{\cos }^2\frac{\mathrm{C}}{2}+(a+b{)}^2{\sin }^2\frac{\mathrm{C}}{2}=c^2$

In any $\Delta ABC$, prove that $\left(b+c-a\right)\left(\cot \frac{B}{2}+\cot \frac{C}{2}\right)=2a\cot \frac{A}{2}$

In $\Delta \mathrm{ABC}$, prove that $(\mathrm{b}+\mathrm{c}-\mathrm{a})\tan \frac{\mathrm{A}}{2}=(\mathrm{c}+\mathrm{a}-\mathrm{b})\tan \frac{\mathrm{B}}{2}=(\mathrm{a}+\mathrm{b}-\mathrm{c})\tan \frac{\mathrm{C}}{2}$.

In $\Delta \mathrm{ABC}$, prove that $\mathrm{a}(\mathrm{cosB}+\mathrm{cosC})=2(\mathrm{b}+\mathrm{c}){\sin }^2\frac{\mathrm{A}}{2}$.