`int 1/(cos(theta) -1) d theta` Find the indefinite integral

Explanation for `cos(x) = (1-u^2)/(1+u^2)`

before that , we know

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

as `cos^2(x)` can be written as `1/(sec^2(x))`

and we can show `sin^2(x) = ((sin^2(x))/(cos^2(x) ))/(1/(cos^2(x)))`

 = `tan^2(x)/sec^2x`

so now ,

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

= `(1/sec^2(x)) - (tan^2(x)/sec^2(x))`

=`(1-tan^2(x))/(sec^2(x))`

but `sec^2(x) = 1+tan^2(x)` ,as its an identity

so,

=`(1-tan^2(x))/(sec^2(x))`

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

so ,

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

so,

then

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

as before we told to assume that `u= tan(x/2),`

so,

`cos(x) = (1-u^2)/(1+u^2)`

Hope this helps to understand better