Question

Show that : 1+ /−1+x = 1+2/²−²

Answer 1

It is required to prove that:

`(\tan x+1)/(\tan x-1)=(1+2\sin x\cos x)/(\sin^2x-\cos^2x)`

Solve the right hand side as follows:

`\begin{gathered} (1+2\sin x\cos x)/(\sin^2x-\cos^2x)=\frac{\sin^2x+\cos^2x+2\sin x\cos x}{(\sin x-\cos x)\cdot(\sin x+\cos x)_{}} \\ =((\sin x+\cos x)^2)/((\sin x-\cos x)\cdot(\sin x+\cos x)) \\ =(\sin x+\cos x)/(\sin x-\cos x) \end{gathered}`

Divide the numerator and denominator by cosx to get:

`\begin{gathered} (\sin x+\cos x)/(\sin x-\cos x)=((\sin x)/(\cos x)+(\cos x)/(cosx))/((\sin x)/(\cos x)-(\cos x)/(cosx)) \\ =\frac{\tan x+1}{\text{tanx}-1} \end{gathered}`

Which is the left side of the equation.

Hence it is proved that:

`(\tan x+1)/(\tan x-1)=(1+2\sin x\cos x)/(\sin^2x-\cos^2x)`