Prove the identity. tan(x+pi/4)=tan(x)+1/1-tan(x)

Answer 1

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

So, tan(x + π/4) = [tanx + tan(π/4)] / [1 - tanxtan(π/4)] = (tanx + 1) / (1 - tanx)

Answer 2

Tangent sum identity:

tan(a+b) = (tan(a)+tan(b))/(1-tan(a)tan(b)

for b = π/4, tan(b) = 1

Therefore:

tan(x+ π/4) = (tan(x)+tan(π/4))/(1-tan(x)tan(π/4)) = (tan(x)+1)/(1-tan(x)(1)) = (tan(x)+1)/(1-tan(x)) QED