Question

In a $△ABC,\frac{r_1+r_2}{1+\cos C}$ is equal to

• A. $\frac{a+b+c}{\Delta }$
• B. $\frac{abc}{2\Delta }$
• C. $\frac{abc}{{\Delta }^2}$
• D. $\frac{2abc}{\Delta }$

Answer 1

The correct option is B $\frac{abc}{2\Delta }$

$\frac{r_1+r_2}{1+\cos C}=4R\frac{\sin \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2}+\sin \frac{B}{2}\cos \frac{A}{2}\cos \frac{C}{2}}{2{\cos }^2\frac{C}{2}}$

$=2R\frac{\sin \frac{A}{2}\cos \frac{B}{2}+\sin \frac{B}{2}\cos \frac{A}{2}}{\cos \frac{C}{2}}$

$=2R\frac{\sin \frac{A+B}{2}}{\cos \frac{C}{2}}$

$=2R\frac{\sin ({90}^{\circ }-\frac{C}{2})}{\cos \frac{C}{2}}$

$=2R=\frac{abc}{2\Delta }$