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adam1231 · Apr 2, 2007, 04:42 AM

cot(x) + tan(x)
1-tan(x) 1-cot(x) = 1+tan(x)+cot(x)

Please verify if you can!!

galactus · Apr 2, 2007, 06:06 PM

There are always many methods to tackle these critters, but I will do a somewhat lengthy one so you can see the identities. OK?

$\frac{cot(x)}{1-tan(x)}+\frac{tan(x)}{1-cot(x)}$

Like any addition, you must make the denominators the same:

Let's multiply the right side by -tan(x):

$\frac{cot(x)}{1-tan(x)}+\frac{tan(x)}{1-cot(x)}\cdot\frac{-tan(x)}{-tan(x)}$

Now you have:

$\frac{cot(x)}{1-tan(x)}-\frac{tan(x)}{1-tan(x)}$

$=\frac{cot(x)-tan^{2}(x)}{1-tan(x)}$

Using $cot(x)=\frac{cos(x)}{sin(x)} \;\ and \;\ tan(x)=\frac{sin(x)}{cos(x)}$ :

We simplify it down to:

$\frac{cos^{2}(x)+sin(x)cos(x)+sin^{2}(x)}{sin(x)co s(x)}$

Since $cos^{2}(x)+sin^{2}(x)=1$

$\frac{1+sin(x)cos(x)}{sin(x)cos(x)}$

So, we're getting there:

$\frac{1}{sin(x)cos(x)}+\frac{sin(x)cos(x)}{sin(x)c os(x)}$

$=\frac{1}{sin(x)cos(x)}+1$

Now, $\frac{1}{sin(x)cos(x)}=cot(x)+tan(x)$

Therefore, we have... drum roll please:

$cot(x)+tan(x)+1$

This is a rather lengthy derivation but if you can follow it, you can follow amore simplistic method.

adam1231 · Apr 2, 2007, 07:42 PM

I appreciate the reply but I still am confused by some of your answer. First I don't see how you derived the new equation when you put it all in terms of sin and cos. Then I don't see how 1/sin(x)cos(x)can be equal to cot(x)+tan(x). If you could elaborate more I would appreciate it.

galactus · Apr 3, 2007, 03:59 AM

That's how you do these things a lot of the time is by converting to sine and cosine.

You must know variuos identities in order to transform the left side into the right side.

$cot(x)=\frac{cos(x)}{sin(x)}$
$tan(x)=\frac{sin(x)}{cos(x)}$

So, we have:

$\frac{cos(x)}{sin(x)}+\frac{sin(x)}{cos(x)}$

Cross multiply:

$\frac{cos^{2}(x)+sin^{2}(x)}{sin(x)cos(x)}$

Remember, $cos^{2}(x)+sin^{2}(x)=1$

Therefore:

$\frac{1}{sin(x)cos(x)}=cot(x)+tan(x)$

luffy · Apr 10, 2007, 08:25 PM

Originally Posted by adam1231

cot(x) + tan(x)
1-tan(x) 1-cot(x) = 1+tan(x)+cot(x)

Please verify if you can!!!!

I have no idea

luffy · Apr 10, 2007, 08:27 PM

I wish I knew who you were. I have mrs.philips as my precal teacher