Question

Prove algebraically that the equation is an identity.
tan(x)+(1/(tan(x)))=1/(sin(x)cos(x))

Answer 1

\[\tan(\theta)+\frac{1}{\tan \theta}=\frac{1}{\sin(\theta)\cos(\theta)}\]
\[\tan(t)+\frac{1}{\tan(t)}=\frac{1}{\sin(t)\cos(t)}\]
they are both the same equation.

Answer 2

tan x + 1/tanx = sinx/cosx + cosx/sinx = sin^2 x + cos^2 x / sinxcosx = 1/sinxcosx

Answer 3

Alright, let's see if I still have my magic touch. Are you ready to work on this with me?

Answer 4

yep still here.

Answer 5

Awesome. First things first: Do you know how else we can show tan? In terms of sin and cosine?

Answer 6

yeah, there are couple of thing. but they are long equations to write out.
him1618 I kind of get it how you got there, but little confused.

Answer 7

Well, there's one universally simple one where you divide something by something to get tan. Do you remember it? You've gotta memorize this one, it's very important.

Answer 8

sinx/cosx i believe.

Answer 9

Perfect! In case you ever forget, I came up with a cheesy way to never forget:

There's an island called "sinkland". Here, the people are called "sinkos", or "sincos". And if you meet a sinco, he will be very TAN.

Regardless, notice what happens when we switch out tan with sin/cos:

Answer 10

sin/cos+(1/((sin/cos))=1/(sin*cos)

Answer 11

Technically you should have x's next to each sin or cos, but I've got a weird thing where I avoid putting them in until I'm all done. It's less messy.

Answer 12

true! i'm getting that habit also.

Answer 13

Does it look good so far?

Answer 14

\[\frac{\sin}{\cos+\frac{1}{\frac{\sin}{\cos}}}=\frac{1}{\sin*\cos}\]
does it look like this?

Answer 15

Actually there should be two fractions on the left side of the equals sign. Can you nudge the (1/sin/cos) to the right of the first fraction?

Answer 16

don't get it

Answer 17

Can you explain to me for a moment how to make fractions appear on this site? Never quite figured that part out.

Answer 18

so it should be (sin/cos)+(1/sin/cos) on the left side?

Answer 19

that's right

Answer 20

\[\frac{\sin}{\cos}+\frac{\frac{1}{\sin}}{\cos}\]

Answer 21

perfect.

Answer 22

Now, one small mathematical "crunch" we will be making. That fraction on the right has two division bars. Do you know the rule about what happens when you do 1/x/y ?

Answer 23

For example, if you had 1/(1/4), or... 1/.25, what would you do?

Answer 24

If it's too tricky, lemme know and I'll show you the easy trick for this.

Answer 25

don't get your question

Answer 26

i kind of get it. but it looks little confusing.

Answer 27

Do you seen on the second fraction we have (1/(sin/cos))? We technically should never have a fraction inside a fraction. So we've gotta simplify it. So if I showed you this fraction: 1/(3/5), what would you do?

Answer 28

If you can show me how to draw fractions here, my examples will make much more sense.

Answer 29

take the 3/5 only. and add +1 i donk know just guessing.

Answer 30

No, I'm afraid not.

Answer 31

i understand your fractions. the complex fractions are fuzzy.
where you click "Post" to the left is "Equation" button/option click on it.

Answer 32

And I just post stuff in the form of 1/4 ?

Answer 33

\[1/4\]

Answer 34

Dang it. :/

Answer 35

Ok, but basically:
Imagine 1 divided by (3 divided by 5)

Normally, the long way to do it would be to take the 1 and convert it to a fraction:

1/1 and multiply it by the reciprocal of the second fraction: 5/3. The final answer would be 5/3

Answer 36

Or... as you can tell, if you ever have 1/(some other fraction), the answer is simply "the other fraction flipped upside down"

Answer 37

ok i get you point. so we ming as well flip that (1/sin/cos) to (cos/sin) which is the same right?

Answer 38

perfect!

Answer 39

Which, if you're bored enough to memorize, is also equal to "cotangent" (the inverse of tangent). You'll want to know that for future trig/calculus stuff.

Answer 40

Alright, so back on topic: we have,

sin/cos + cos/sin = 1/cossin

Answer 41

at some point i'm going to have to know all of them.

Answer 42

Erm... lemme reword that:
sin/cos + cos/sin = 1/cos*sin

Answer 43

(sin/cos)+(cos/sin)=1/(cos*sin) ???

Answer 44

perfect. Now, on the left side, we can just cross cancel, right?

Answer 45

because there's a sin on the top of one and a sin on the bottom of the other, right?

Answer 46

hit equation on the bottom of your typing screen. and to write a fraction type
frac{1}{3} for (1/3)

Answer 47

Awesome! Thanks.

Answer 48

\[\frac{\sin}{\cos} + \frac{\cos}{\sin} = \frac{1}{\cos*\sin}\]

Answer 49

Alright, so can we cancel out the sins and cos to get all ones?

Answer 50

what are you trying to get me here to? because i'm looking at my question and your explanation and how to make sense of either. it kind of confuses me.

Answer 51

Ok, that's a good thing then. Because it was a trick question. :P

Answer 52

and yes with your above equation you cancel them out the get ones

Answer 53

nononono! There's a plus sign. You can only cross cancel if you're multiplying them together.

Answer 54

I was using that trick question to make sure you don't accidentally do that on a test. ^_^

Answer 55

you substitute in one side of the equation for the other side and manipulate it.

Answer 56

and the other way around.

Answer 57

No need to do that yet. We're going to make the left side equal 1/(cos*sin) in a few moments.

Answer 58

Now, how do we add \[\frac{\sin}{\cos} + \frac{\cos}{\sin}\] ?

Answer 59

beeeehhhiiiivvvveeee

Answer 60

what do you mean how do we add?

Answer 61

that equals 1/(sin*cos)

Answer 62

Eventually it will, yes. But how do we add the fractions?

Answer 63

As in, what is the rule for adding any two fractions?

Answer 64

make the denominators alike. so multiply the left denominator by right side. and right denominator by left side of the plus sign

Answer 65

perfect.

Answer 66

So what do we multiply \[\frac{\sin}{\cos}\] by to match the fraction next to it?

Answer 67

Or at least to eventually match the fraction next to it?

Answer 68

sin

Answer 69

which would equal in case you are going to ask
\[\frac{\sin^2+\cos^2}{\sin*\cos}\]

Answer 70

yeppers.

Answer 71

Now, here's where stuff gets tricky if you don't have your identities memorized.

Answer 72

Do you know of any pythagorean identities?

Answer 73

hint hint: http://www.sosmath.com/trig/Trig5/trig5/trig5.html

Answer 74

not memorized. but i've got them written down.
by the way is this on problem or extra Info info?

Answer 75

it's for this problem, I'm afraid.

Answer 76

cool

Answer 77

But have no fear! Google will rescue us valiantly.

Answer 78

http://www.sosmath.com/trig/Trig5/trig5/trig5.html

Answer 79

Specifically part two, pythagorean identities. Which one applies?

Answer 80

sin^2+cos^2=1

Answer 81

awesome. And what do we get?

Answer 82

1

Answer 83

BTW, look at what our buddy iman posted on the new questions list. Hee hee.

Answer 84

Well, I mean after we swap out the identity in our left side problem?

Answer 85

yeah i've noticed

Answer 86

And you also noticed what happened to our left hand side equation?

Answer 87

tan+(1/tan) ???

Answer 88

Wait. Go back to the last thing we had.

Answer 89

(sin^2+cos^2) / (sin∗cos)

Answer 90

\[\frac{\sin^2+\cos^2}{\sin∗\cos}\]

Answer 91

Remember, you used your identity to swap out your parentheses with a 1, right?

Answer 92

1/(sin*cos)

Answer 93

great!

Answer 94

What is our goal?

Answer 95

1/(sin*cos)

Answer 96

Feel accomplished? ;)

Answer 97

nope.
tan+1/tan
with what do we substitute that out? i'm confused here.

Answer 98

No, we did that all already!