Question

Tan x/(1+tan x) = ? Or prove that : tan x/(1+tan x) = sin x.cos x

Answer 1

Please refer to the Explanation.

Explanation:

This can not be proved, because it is a false statement.

If it were true, then it must be so for `x=pi/3`.

But, for `x=pi/3`, we find that,

`"The L.H.S."=tan(pi/3)/(1+tan(pi/3))=sqrt3/(1+sqrt3), and,`,

`"The R.H.S."=sin(pi/3)cos(pi/3)=sqrt3/2*1/2=sqrt3/4`.

Clearly, `"The L.H.S."!="The R.H.S."`

In fact, `tanx/(1+tan^2x)=sinxcosx`, as proved below :

Using the Identity : `1+tan^2x=sec^2x`, we have,

`tanx/(1+tan^2x)`,

`=tanx/sec^2x=tanx*cos^2x=sinx/cosx*cos^2x`.

`rArr tanx/(1+tan^2x)=sinxcosx`.