Question

Select the expression below that is identical to \[\frac{ 1 }{ 1-\cos x }+\frac{ 1 }{ 1+\cos x }\]

A.)\[2\sec ^{2}\theta\]
B.)1
C.)\[2\cos ^{2} x\]
D.)\[2\sin ^{2} x\]
E.)\[2\csc ^{2} x\]

Answer 1

Do you know how to make a common denominator so you can add those two fractions together? After you do so, I think you'll see something you recognize from your trig identities...

Answer 2

yes. But i'm not sure I can handle the mess that ensues.

Answer 3

Sure you can...

Answer 4

ALL RIGHT. Lets give it a try! :)

Answer 5

Here, let me suggest a substitution that will make it less scary-looking :-)
Let \(a = \cos x\) so we can rewrite it as
\[\frac{1}{1-a}+\frac{1}{1+a}\]Now we multiply the left fraction top and bottom by the denominator of the right fraction, and the right fraction top and bottom by the denominator of the left fraction. Do that and tell me what you get

Answer 6

\[\frac{ 1(1+\cos x) }{ (1-\cos x)(1+\cos x) }+\frac{ 1(1- \cos x) }{ (1-\cos x)(1+\cos x) }\]

Answer 7

I already started before you suggested the "a" idea.

Answer 8

So now you just add then simplify.

Answer 9

Right. Either before or after you add, I'd like you to multiply out the denominator...

Answer 10

what do you mean?

Answer 11

You've got (1-a)(1+a) — what is that after you expand it?

Answer 12

(1-cos x)(1+cos x) if you prefer

Answer 13

1 + cos x ^2

Answer 14

close, isn't it 1 - cos^2 x?

Answer 15

yes. Shame upon my family. :)

Answer 16

So thats the both the bottom and the top portions
so it equals one.

Answer 17

it happens. easier to see what you're doing with my trick, which is why I like it.

sin^x + cos^x = 1
1 - cos^x = sin^x
(that's the trig identity I was hoping you would remember)

so what does the numerator simplify to?

Answer 18

okay okay so the denominator = sin^2, the numerator would be 1(1+a)+1(1-a) which is just (1+a)+(1-a)

Answer 19

and that simplifies to...

Answer 20

1

Answer 21

If i'm not retarded which is possible so...

Answer 22

1 + a + 1 - a = ?

Answer 23

we could rearrange the order to 1 + 1 + a - a = 1 + 1 = 2

Answer 24

Here i was overthinking the whole thing. Thats embarrassing.

Answer 25

okay so its 2/ sin ^2

Answer 26

so, that gives us
\[\frac{2}{sin^2 x}\]

but that isn't on our list of choices, is it?

Answer 27

nope.

Answer 28

they throw in 2 sin^x on the list as a trap for the unwary, but we're too smart for that!

do you remember the definition of secant and cosecant?
(sec x, csc x)

Answer 29

one is the reciprocal of sin x, the other is the reciprocal of cos x. boy, wouldn't it be handy if they had matching first letters to help you remember which is which :-) they don't, however.

Answer 30

\[\sec x = \frac{1}{\cos x}\]\[\csc x = \frac{1}{\sin x}\]

Answer 31

so we have 2 csc ^2 x which is answer E

Answer 32

I just need like a super list of all the identities.

Answer 33

You were awesome dude, thanks for helping me out. I actually understood it :)

Answer 34

you got it! yeah, a lot of it is just practice, and learning to recognize the identities in different forms...

Answer 35

I'll confess I had to look up whether sec or csc was the one we wanted :-)

Answer 36

anyhow, remember my substitution trick. it works well with expressions that have lots of square roots and so on in them, too. if nothing else, you save some time writing :-)

Answer 37

Okay sounds like a deal. :) I'll try to figure out the next one on my own before posting it so expect me back soon! :)

Answer 38

for extra speed, remember that whenever you have fractions like you have in this one where the denominators are x-y and x +y that those are the factors of x^2 -y^2 and you can just right there

Answer 39

thats pretty sweet. thanks! You're just stuffed with math tips. I'm more of a writer so I don't get this math stuff. :)