Question

If `A+B+C=pi`, prove that : `tan A/2 tan B/2 + tan B/2 tan C/2 + tan C/2 tan A/2 =1`

Answer 1

Here , `A+B+C=pi`
`implies((A)/(2)+(B)/(2))=((pi)/(2)-( C)/(2))`
`implies tan ((A)/(2)+(B)/(2))=tan ((pi)/(2)-( C)/(2))=cot ""(C)/(2)`
`implies (tan ""(A)/(2)+tan ""(B)/(2))/(1-tan""(A)/(2)tan""(B)/(2))=(1)/(tan""( C)/(2))`
`implies tan ""(C)/(2)tan ""(A)/(2)+tan ""(B)/(2)tan ""( C)/(2)=1-tan""(A)/(2) tan""(B)/(2)`
`implies tan""(A)/(2)tan ""(B)/(2)+tan ""(B)/(2)tan ""( C)/(2)+tan""(C )/(2)tan""(A)/(2)+1.`