Question

In a $\Delta ABC,thevalueof(\frac{a}{\tan A}+\frac{b}{\tan B}+\frac{c}{\tan C})$ is equal to

• A. $2r$
• B. $r+2R$
• C. $2r+R$
• D. $2(r+R)$

Answer 1

The correct option is D $2(r+R)$

$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R$

Now,

$\frac{a}{\tan A}=\frac{b}{\tan B}=\frac{c}{\tan C}$

$\frac{a}{\sin A}.\cos A+\frac{b}{\sin B}.\cos B+\frac{c}{\sin C}.\cos C$

$=2R(\cos A+\cos B+\cos C)$

$=2R(1+4\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2})$

$=2R[1+\frac{r}{R}]$

$=2(R+r)$ is the answer