यदि ` x+ y + z= xyz`, तो त्रिकोणमिति द्वारा सिद्ध कीजिए कि -
` (2x)/(1-x^(2)) + (2y)/(1-y^(2))+ (2z)/(1-z^(2))= (2x)/(1-x^(2))* (2y)/(1-x^(2))*(2y)/(1-y^(2)) * (2z)/(1-z^(2))`

Explanation

माना ` x = tan A, y = tan B, z = tan C`
अब ` x+ y+z = xyz`
` rArr x+y = -z + xyz = -z (1- xy)`
`(sin B - sin C)/(sin B + sinC) * cot. A/2 `
` = (2 cos. (B+C)/2 sin. (B-C)/2 . cot. A/2)/(2 sin . (B+C)/2 cos. (B-C)/2 ).cot. A/2`
` = cot (90^(@) - A/2) * tan. (B-C)/2 .cot. A/2 `
` ( :' A + B+ C = 180^(@))`
` = tan . A/2 * tan . (B-C)/2 * 1/(tan.A/2)`
अब ` tan. (B-C)/2 = (b-c)/(b+c) * cot. A/2`
इसी प्रकार सिद्ध किया जा सकता है कि
` tan (A-B)/2 = (a-b)/(a+b) cot. C/2 `