trig func to alg expression confusion

[Kuroki]

Example given and steps:

1. sin(arctan(u))

2. sin(theta)

3. tan(theta)cos(theta)

4. tan(theta)
sec(theta)

5. tan(theta)
sqrt(1+tan^2(theta))

6. u
sqrt(1+u^2)

I understand how you get from most steps to the next except for probably the most important part: where did arctan go from the first to the second step
And where did the sec(theta) come from in step 4?

 

[Deleted member 4993]

Example given and steps:

1.

2. sin(theta)

3. tan(theta)cos(theta) = \(\displaystyle \ tan(\theta) \ * \ \dfrac{1}{\frac{1}{cos(\theta)}}\)

4. tan(theta)
sec(theta)

5. tan(theta)
sqrt(1+tan^2(theta))

6. u
sqrt(1+u^2)

I understand how you get from most steps to the next except for probably the most important part: where did arctan go from the first to the second step
And where did the sec(theta) come from in step 4?

As displayed, your step 1 is empty

 

[Kuroki]

edited

Sorry, about that. It was showing up earlier, so that's weird.
1. sin(arctan(u))

 

[mmm4444bot]

[The image] was showing up earlier

sin(arctan(u))

:idea: There is no need to post an image for an expression so easy to type.

the most important part: where did arctan go from the first to the second step

The arctan function outputs an angle. Therefore, the symbol arctan(u) represents an angle.

In step two, they simply switched to another symbol for the angle. In other words, they defined:

arctan(u) = theta

 

[mmm4444bot]

where did the sec(theta) come from in step 4?

Subhotosh already answered this question, with his red crayon.