Question

if A+B+C=180 then prove that
tanA+tanB+tanC=tanA×tanB×tanC​

Answer 1

To prove--->

If A + B + C = 180°

then

tanA + tanB + tanC = tanA tanB tanC

Proof ---> A + B + C = 180° ( given )

A + B = 180° - C

Taking tan both sides

tan ( A + B ) = tan ( 180° - C )

We have some formulee

tan x + tan y

tan ( x + y ) = ------------------------

1 - tan x tan y

and tan (180° - θ ) = - tan θ

Applying these formulee here

tan A + tan B

---------------------- = - tanC

1 - tanA tan B

=> tan A + tan B = -tanC( 1- tanA tanB )

=> tanA + tanB = -tanC + tanA tanB tanC

=> tanA + tanB + tanC = tanA tanB tanC

Hence proved

Additional information--->

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

2)Sin(A + B ) = SinA CosB + CosA SinB

3)Sin(A - B ) = SinA CosB - CosA SinB

4)Cos(A +B )= CosA CosB - SinA SinB

5)Cos(A - B )= CosA CosB + SinA SinB