Understanding the Simplification of ((1 - cos A)/ (1+cos A) ) ^(1/2)

[Miike012]

Homework Statement

Prove:( (1 - cos A)/ (1+cos A) ) ^(1/2) = cosec A - cot A

Then they have...

((1 - cos A)/ (1+cos A) ) ^(1/2) = ( (1 - cos A)^2 )/ (1-cos^2 A) ) ^(1/2)

= (1 - cos A) / (1-cos^2 A) ^(1/2)/

I have never done anything like this... I have just been studying trig for the past two days...

Why did they square the entire numerator but only square the cos in the denominator?

... Can some help me understand this the easiest way possible? thank you.

Homework Equations

The Attempt at a Solution

 

[Miike012]

Nope lol sorry... However I did figure it out on my own by just looking at it! So I was proud about that lol...

One important question that I would like answered...
Lets say... I have... (1 - cos A)/ (1+cos A) ) ^(1/2)
Should I leave it as so... or try and simplify it down to cosec A - cot A?
Because to be honest I don't think I could have looked at the square root portion and though " Ohh... maybe this can be simplified..."

 

[HallsofIvy]

Homework Statement

Prove:( (1 - cos A)/ (1+cos A) ) ^(1/2) = cosec A - cot A

Then they have...

((1 - cos A)/ (1+cos A) ) ^(1/2) = ( (1 - cos A)^2 )/ (1-cos^2 A) ) ^(1/2)

= (1 - cos A) / (1-cos^2 A) ^(1/2)

Once you are here, use the fact that 1- cos^2(A)= sin^2(A)
(From sin^2(A)+ cos^2(A)= 1. Do you know that identity?)

\left(\frac{1- cos(A)}{1- cos^2(A)}\right)^{1/2}= \frac{1- cos(A)}{sin(A)}= \frac{1}{sin(A)}- \frac{cos(A)}{sin(A)}
Do you know the definition of "cosec(A)" and "cot(A)"?

I have never done anything like this... I have just been studying trig for the past two days...

Why did they square the entire numerator but only square the cos in the denominator?

They didn't. What they did is multiply both numerator and denominator by 1- cos(A). Since there was already 1- cos(A), that becomes (1- cos(A))^2. The denominator was 1+ cos(A) so it becomes (1- cos(A))(1+ cos(A))= 1+ 1(cos(A))- cos(A)(1)- cos(A)cos(A)= 1- cos^2(A) because the "1(cos(A))"and "[mat]-cos(A)(1)[/itex] cancel.

... Can some help me understand this the easiest way possible? thank you.

Homework Equations

The Attempt at a Solution

 

[Miike012]

Ive been reading in my book.. and it says,

sin^2 A + cos^2 A = 1
Sec^2A = 1 + tan^2A
.
.
. Should I also rearange sin^2 A + cos^2 A = 1 ; cos^2 A = 1 -sin^2 A = sin^2 A =1 -cos^2..
and the same with the other one?
Should I not only memorize the two but also memorize their rearangements?