Question

Establish the identity: sec(u)-tan(u)=cos(u)/1+sin(u)

Answer 1

Okay, so we obviously begin with

\[\frac{1}{\cos(u)} - \frac{\sin(u)}{\cos(u)} = \frac{1 - \sin(u)}{\cos(u)}\] right?

Answer 2

Yes. Please lead me from there.

Answer 3

Basically, we know that we need \(1 + \sin(u)\) in the denominator, so what we can do is multiply \(\dfrac{1- \sin(u)}{\cos(u)}\) by \(\dfrac{1 + \sin(u)}{1 + \sin(u)}\)

Answer 4

When we do that we'll have

\(\dfrac{1- \sin(u)}{\cos(u)} \times \dfrac{1 + \sin(u)}{1 + \sin(u)} = \dfrac{1 - \sin^2(u)}{\cos(u)(1 + \sin(u))} \)

Answer 5

Do you remember what \(1 - \sin^2(u)\) is equivalent to?

Answer 6

If I may ask, are you allowed to multiply the denominator of the RHS by the LHS. My instructor has only taught me to work strictly with one side to prove the other side.

Answer 7

We ARE working with the LHS. I multiplied top and bottom by 1 which is legal.

Answer 8

Okay, thank you for explaning.

Answer 9

I assure you that this is the only way to do this. There's no other way to get to the end result.

Answer 10

Basically, we're allowed to multiply by 1. Also we can make 1 equal to whatever fraction we want as long as the expression at the top is the same as the expression in the denominator. That step is 100% legal.

Answer 11

Okay, thank you so much for assisting me with this problem. I really appreciate it.

Answer 12

I suppose you can figure out the rest from here?

Answer 13

Yes, thank you.

Answer 14

You're welcome