Hi all.
I'm trying to show that...
1 - ( tan (x/2) )^2 = sec x + tan x
Any help will be appreciated.
Thanks.
OK. Let me start from the start...
I was working on the indefinite integral of sec x...
INT (sec x) dx...
I did this two different ways:
Method 1:
sec x * [ (sec x + tan x) / (sec x + tan x) ]
.
.
.
leads to a simple substitution t = sec x + tan x...
and the integral evaluates to...
ln | sec x + tan x | + c
Method 2:
Using Weierstrass Substitution...u = tan (x/2)etc etc...
.
.
.
I end up with the answer ln | 1 - ( tan (x/2) )^2 | + c
Now I'm trying to prove that both these answers (methods 1 and 2) are the same, hence the earlier post.
Where did I go wrong?
Firstly, this is not right! Check your calculations. Secondly, the two integrals don't have to be equal. In the first substitution you have found that . What does the indicate? If, say, we had some function and when we integrate, we find some anti-derivative . Now, if for all the anti-derivatives we can find, , it's true that , for what purpose would there be for to serve? We use precisely because this is not the case! Consider the integral you have found in the first substitution satisfies your , for ANY constant . is valid for and so is . So what, for example, if the other substitution gives you , for some constant ? is a constant, and what we donated by was a constant, so both satisfy . In general, any two or more anti-derivatives of some function F(x) don't have to be equal; rather, they have to differ by a constant.