TRIG SUBSTITUTION INTEGRAL- Where did I go wrong?

[funkinvile]

< link to objectionable page removed >

I used this substitution:

\(\displaystyle x\, =\, \tan(\theta)\)

\(\displaystyle dx\, =\, \sec^2(\theta)\, d\theta\)

Then I did:

\(\displaystyle \displaystyle \int\, \dfrac{1}{x^2\, \sqrt{\strut\, 1\, +\, x^2\,}}\, dx\, =\, \int\, \dfrac{\sec^2(\theta)}{x^2\, \sqrt{\strut\, 1\, +\, \tan^2(\theta)\,}}\, d\theta\, =\, \int\, \dfrac{\sec^2(\theta)}{x^2\, \sqrt{\strut\, \sec^2(\theta)\,}}\, d\theta\, =\, \int\, \dfrac{\sec^2(\theta)}{\tan^2(\theta)\, \sec(\theta)}\, d\theta\)

\(\displaystyle \displaystyle =\, \int\, \dfrac{\sec(\theta)}{\tan^2(\theta)}\, d\theta\, =\, \int\, \dfrac{\left(\dfrac{1}{\cos(\theta)}\right)}{\left(\dfrac{\sin(\theta)}{\cos(\theta)}\right)}\, d\theta\, =\, \int\, \left(\dfrac{1}{\cos(\theta)}\right)\, \left(\dfrac{\cos(\theta)}{\sin(\theta)}\right)\, d\theta\, =\, \int\, \csc(\theta)\, d\theta\)

On this <reference removed>, I have a picture of my paper work and a picture of the solution manual to the problem I am trying to solve.
My question is: how is it that the solutions manual uses "1/cos(x)" and ends up with the integral of "cos(x)/sin^2(x)" whereas when I use "sec(x)" and do the math I end up with the integral of "csc(x)"?

Please tell me where I went wrong I am so dang frustrated

 

[stapel]

\(\displaystyle \displaystyle \int\, \dfrac{1}{x^2\, \sqrt{\strut\, 1\, +\, x^2\,}}\, dx\, =\, \int\, \dfrac{\sec^2(\theta)}{x^2\, \sqrt{\strut\, 1\, +\, \tan^2(\theta)\,}}\, d\theta\, =\, \int\, \dfrac{\sec^2(\theta)}{x^2\, \sqrt{\strut\, \sec^2(\theta)\,}}\, d\theta\, =\, \int\, \dfrac{\sec^2(\theta)}{\tan^2(\theta)\, \sec(\theta)}\, d\theta\)

\(\displaystyle \displaystyle =\, \int\, \dfrac{\sec(\theta)}{\tan^2(\theta)}\, d\theta\, =\, \int\, \dfrac{\left(\dfrac{1}{\cos(\theta)}\right)}{\left(\dfrac{\sin(\theta)}{\cos(\theta)}\right)}\, d\theta\)

What happened to the square on the tangent? Why are the sine and cosine generated by that squared tangent not themselves squared?

 

[funkinvile]

What happened to the square on the tangent? Why are the sine and cosine generated by that squared tangent not themselves squared?

Ahh thank you! Can't believe I didn't see that very much appreciated.