Question

If A + B = C = 180°, then the value of ${\displaystyle \cot \frac{A}{2}+\cot \frac{B}{2}+\cot \frac{C}{2}}$ will be ______.

विकल्प

• ${\displaystyle 2\cot \frac{A}{2}\cot \frac{B}{2}\cot \frac{C}{2}}$

• ${\displaystyle 4\cot \frac{A}{2}\cot \frac{B}{2}\cot \frac{C}{2}}$

• ${\displaystyle \cot \frac{A}{2}\cot \frac{B}{2}\cot \frac{C}{2}}$

• ${\displaystyle 8\cot \frac{A}{2}\cot \frac{B}{2}\cot \frac{C}{2}}$

Answer 1

If A + B = C = 180°, then the value of ${\displaystyle \cot \frac{A}{2}+\cot \frac{B}{2}+\cot \frac{C}{2}}$ will be ${\displaystyle \underline{\cot \frac{A}{2}\cot \frac{B}{2}\cot \frac{C}{2}}}$.

Explanation:

A + B + C = 180°

∴ ${\displaystyle \frac{A}{2}+\frac{B}{2}={90}^{\circ }-\frac{C}{2}}$

∴ ${\displaystyle \cot (\frac{A}{2}+\frac{B}{2})=\cot ({90}^{\circ }-\frac{C}{2})}$

or ${\displaystyle \frac{\cot \frac{A}{2}.\cot \frac{B}{2}-1}{\cot \frac{B}{2}+\cot \frac{A}{2}}=\tan \frac{C}{2}=\frac{1}{\cot \frac{C}{2}}}$

or ${\displaystyle (\cot \frac{A}{2}\cot \frac{B}{2}-1)\cot \frac{C}{2}=\cot \frac{B}{2}+\cot \frac{A}{2}+\cot \frac{A}{2}.\cot \frac{B}{2}.\cot \frac{C}{2}}$

= ${\displaystyle \cot \frac{C}{2}+\cot \frac{B}{2}+\cot \frac{A}{2}}$