Question

Hi! I need help with this practice problem. My teacher gave us the answer, "2cot u" BUT she wants us to figure out why that's the correct answer. I'm having some trouble with this and could really use some help. Thank you!

Problem: 1+ sec u/tan u - tan u/1+sec u=

Answer 1

You need brackets:
(1+ sec u)/tan u - tan u/(1+sec u)

I am going to guess that you have just learned:
sin^2 x + cos^2 x = 1
and its variations ...
tan^2 x + 1 = sec^2 x
cot^2 x + 1 = csc^2 x

So
(1+ sec u)/tan u - tan u/(1+sec u)
= ( (1+secu)^2 - tan^2 u)/(tanu(1+secu))
= (1 + 2secu + sec^2 u - tan^2 u)/(tanu(1+secu)
= (1 + 2secu + tan^2 u + 1 - tan^2 u)/(tanu(1+secu))
= (2 + 2secu)/(tanu(1+secu))
= 2(1 + secu)/(tanu(1+secu))
= 2/tanu
= 2cotu