How do you simplify (1+ tanx) / (1- tanx)?

Solution

Consider

\(\frac{1+tan x}{1-tan x}\)………………………..(1)

We know that

\(tan(A+B)=\frac{tanA + tanB}{1-tanAtanB}\)

and also

\(tan(\frac{\Pi }{4})= 1\)

Hence equation (1) becomes

\(\frac{tan \frac{\pi }{4} + tan x}{1-tan\frac{\pi }{4} tanx}\)

since \(tan\left ( \frac{\pi}{4} \right ) = 1\)

⇒ \(tan\left ( x +\frac{\pi}{4} \right )\)

∴  \(\frac{1+tanx}{1-tanx}= tan\left ( x+\frac{\pi}{4} \right )\)

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