Question

Prove that sec(theta) / tan(theta) = csc(theta)
please explain!

note: there should be 3 steps

Answer 1

Do you know what sec(theta) and tan(theta) can be replaced with?

Answer 2

sec(theta) = 1/cos(theta) and tan(theta)= 1/cot(theta)

Answer 3

Right. But the one you replaced tan with isn't going to be quite as useful. Do you know what else is the same as tan(theta)?

Answer 4

sin(theta)/cos(theta) ?

Answer 5

There ya go. So now we would have:
\[\frac{ 1 }{ \cos \theta } \div \frac{ \sin \theta }{ \cos \theta }\]
Think you can see what will happen from here?

Answer 6

oh yeah! i got it thank you!

Answer 7

Awesome :3

Answer 8

do you think you can helo me with this one too?
solve -sin^2 x= 2cosx - 2

Answer 9

Solve for x or prove?

Answer 10

solve please

Answer 11

Ah, alrighty. Lemme look.

Answer 12

Mkay, so the things to recognize is that we have a squared term and first power term. I see that and I think maybe its a quadratic. If it is, then we need to get everything to be sin or everything to be cos, as well as make sure they have the same angle. Well, they already have the same angle, so now we just need them to be the same trig function. Do you know how you might substitute sin^2(x) into something with cosines?

Answer 13

sin^2 x= (1-cos2x) / 2

Answer 14

Yes, but that gives us different angles. We want them all to just be x, not 2x. It's good you know that identity, though xD Well, we need to use this identity to help us get there:

\[\sin ^{2}x + \cos ^{2}x = 1\] and then we solve for sin^2(x)

Answer 15

so sin^2x = 1- cos^2x ?

Answer 16

Yep, bingo. So this would give us:

\[-(1-\cos ^{2}x) =2cosx - 2\]

So now we have all we need to make this into a quadratic factoring equation, now we just need to set evertything equal to 0 and factor.

Answer 17

yeah that's the part i have trouble with.. i'm not quite sure how to make it into a quadratic function and then factor it

Answer 18

Did we at least set it up properly like a quadratic factoring problem?

Answer 19

No i think we have to have one side to = 0

Answer 20

No, I was just curious if you had doen that yourself yet or not xD

Answer 21

\[\cos ^{2}x - 2cosx +1 = 0\]
Sorry bout that :P

Answer 22

Alright, so what I'm going to do is pretend cosx is simply just x.
\[x ^{2} -2x +1 = 0\]
Now we factor that.

Answer 23

so (x-1) (x-1)

Answer 24

Correct. So x = 1. Now we just re-replace x with cosx to get cosx = 1

Answer 25

so that means x =0. Is that my answer or should i leave like cosx=1 ?

Answer 26

No, x = 0 is the answer xD

Answer 27

Okay thank you so much for all your help, i really appreciate it! :)

Answer 28

Np :3