Question

In a triangle ABC with usual notations, if a,b, and c are in arithmetic progression, then, \[\tan\frac{A}{2}\cdot\tan\frac{C}{2}=\]

Options

• 3

• \[\frac{1}{13}\]

• -3

• \[\frac{1}{3}\]

Answer 1

\[\frac{1}{3}\]

Explanation:

Since a, b, c are in A.P.,

a + c = 2b

\[\mathrm{s}=\frac{\mathrm{a}+\mathrm{b}+\mathrm{c}}{2}=\frac{3\mathrm{b}}{2}\]

\[\tan\frac{\mathrm{A}}{2}\mathrm{tan}\frac{\mathrm{C}}{2}=\sqrt{\frac{(\mathrm{s-b})(\mathrm{s-c})}{\mathrm{s}(\mathrm{s-a})}}\sqrt{\frac{(\mathrm{s-a})(\mathrm{s-b})}{\mathrm{s}(\mathrm{s-c})}}\]

\[=\frac{\mathrm{s-b}}{\mathrm{s}}=\frac{\frac{3\mathrm{b}}{2}-\mathrm{b}}{\frac{3\mathrm{b}}{2}}\]

\[=\frac{1}{3}\]