किसी त्रिभुज ABC में यदि `A = tan^(-1)2` तथा `B = tan^(-1)3` सिद्ध करे कि `C = (pi)/(4)`.

Explanation

दिया है, `A = tan^(-1) 2 rArr tanA = 2` तथा `B = tan^(-1)3 rArr tan B = 3`
चूँकि A,B,C त्रिभुज के कोण है, `therefore A + B + C = pi`
`therefore C = pi - (A+B) " "...(1)`
अब, `A + B = tan^(-1)3 = pi +tan^(-1)[(2+3)/(1-2.3)]`
`[because tan^(-1) x+tan^(-1)y = pi + tan^(-1)[(x+y)/(1-xy)]`
for `x gt 0, y gt 0` तथा `xy gt 1` यहाँ `xy = 2*3 = 6 gt1]`
` = pi + tan^(-1) (-1) = pi - tan^(-1) (1) = pi - (pi)/(4) = (3pi)/(4)`
(1) से, `C = pi - (3pi)/(4)l = (pi)/(4)`