Prove `(sinx+sin(x/2))/(1+cosx+cos(x/2))=tg(x/2)` .

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We will need the following 3 formulas:

`sin2x=2sinxcosx` if we put `x/2`  ` ` instead of `x` we get

`sinx=2sin (x/2) cos (x/2)`                                        (1)      

`cos2x=cos^2x-sin^2x`  if we put `x/2` instead of `x` we get

`cos x=cos^2(x/2)-sin^2(x/2)`                                    (2)

`sin^2x+cos^2x=1`  if we put `x/2` instead of `x` we get

`sin^2(x/2)+cos^2(x/2)=1`                                          (3)

Now let's start from the left side and thry to get the right hand side.


Now we use formula (1) for numerator and formula (2) for denominator.


Now we use formula (3) for `1` in denominator.



and since `tan x=(sinx)/(cosx)` we get


which proves your identity.

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