How Does U-Substitution Using Tan(x/2) Simplify Trigonometric Integrals?

[MrOranges]

Homework Statement

Im trying to figure out how to get sinx=2u/(u^2+1) and cosx=(1-u^2)/(u^2+1). In order to solve the problem integral sinx/(sinx+cosx)

Homework Equations

use tan(x/2) to solve the problem.

The Attempt at a Solution

all I can get to is u=sin(x/2)/cos(x/2)-------2u=sinx/cosx
I don't understand how we can get the u^2+1 on the bottom. All we need is the give the u-substitution but i donno how to do it.

 

[micromass]

So, we let u=\tan(x/2). Then we need to evaluate

\frac{2u}{1+u^2}=\frac{2\tan(x/2)}{1+\tan^2(x/2)}

Try to prove the following formula first:

1+\tan^2(\alpha)=\frac{1}{\cos^2(\alpha)}

 

[MrOranges]

I just want to know how you get sinx=2u/(u^2+1) and cosx=(1-u^2)/(u^2+1) from tan(x/2). After that I can use that to plug in the integral sinx/(sinx+cosx) to make the equation solvable.

 

[micromass]

I don't quite get it. Just substitute u=\tan(x/2) and you'll see that the equation is correct...

 

[MrOranges]

thanks, i got it now

 

[AfterSunShine]

Hello,

Another easier approach :

You have u=tan(x/2), draw the triangle.

Now, sin(x)=sin( 2 (x/2) ) = 2 sin(x/2) cos(x/2) , get sin(x/2) & cos(x/2) from your triangle.
cos(x)=cos( 2 (x/2) ) = [ cos(x/2) ]^2 - [ sin(x/2) ]^2