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=

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

No problem! I can definitely help you understand why the answer is "2cot u" for the given problem.

To solve this, we'll start by simplifying each individual term of the expression.

1. Simplifying 1 + sec u / tan u:
To combine 1 and sec u / tan u, we need to find their common denominator. The common denominator of 1 and tan u is 1, so we can rewrite 1 as (tan u / tan u) and combine the terms:
(1 + sec u) / tan u = (tan u / tan u + sec u) / tan u = (tan u + sec u) / tan u.

2. Simplifying tan u / (1 + sec u):
To combine tan u and 1 + sec u, we need to find their common denominator. The common denominator of tan u and sec u is sec u, so we can rewrite 1 as (sec u / sec u) and combine the terms:
tan u / (1 + sec u) = tan u / (sec u / sec u + sec u) = tan u / sec u = tan u * (1 / sec u) = tan u / (1/cos u) = tan u * cos u.

Now we can simplify the expression by plugging in the above results:

(tan u + sec u) / tan u - (tan u * cos u)

Next, let's simplify each term individually:

1. Simplify (tan u + sec u) / tan u:
To simplify this expression, we need to find a common denominator. The common denominator for tan u and sec u is tan u. We can rewrite tan u as (tan u / 1) to have a common denominator:
(tan u + sec u) / tan u = (tan u / 1 + sec u) / tan u = (tan^2 u + sec u) / tan u = (sin^2 u / cos^2 u + 1 / cos u) / (sin u / cos u) = (sin^2 u + cos u) / (sin u * cos u).

2. Simplify tan u * cos u:
Using the trigonometric identity that tan u = sin u / cos u, we can rewrite tan u * cos u as sin u.

Now we can substitute these simplified expressions back into the original problem:

(sin^2 u + cos u) / (sin u * cos u) - sin u

Next, we need to simplify the expression further:

(sin^2 u + cos u) - sin u * (sin u * cos u) / (sin u * cos u)
= sin^2 u + cos u - sin^2 u
= cos u.

Finally, we have cos u as the simplified expression, which means that the original expression can be simplified to just cos u.

Therefore, the answer is "cos u" and not "2cot u," as your teacher suggested. It seems that your teacher might have made a mistake in their answer.