In a triangle ABC if A B C verify the following identities

In a triangle ABC, if A + B = C, verify the following identities. a) tan A tan B tan c identicalto = tan C - tan A - tan B b sin A + sin (B + C) identicalto 2sin C cos B

Solution

tan(A)tanBtanC = tanC - tanA- tanB

LHS we know tan(A+B) = [tanA + tanB]/[1- tanAtanB]

So, 1- tanAtanB = [tanA+ tanB)/tan(A+B)

tanAtanB = 1 - [ tanA + tanB]/tan(A+B)

tanAtanB = 1 - [tanA +tanB]/tanC

= [tanC - tanA - tanB]/tanC

So, tanAtanBtanC = tanC - tanA - tanB

Hence proved

b) sinA + sin(B+C) = 2sinCcosB

A = C- B

So, sin(C - B) + sin(B +C)

sinCcosB - cosCsinB + sinBcosC+sinCcosB

2sinCcosB

RHS