Prove the Trig Identity

$\displaystyle Tan\;\theta\;+Cot\;\theta\;=\;Csc\;\theta\;Sec\;\t heta$

$\displaystyle \frac{Sin\;\theta}{Cos\;\theta}\;+\;\frac{Cos\;\th eta}{Sin\;\theta}\;=\;Csc\;\theta\; Sec\;\theta$

$\displaystyle \frac{Sin^2\;\theta\;+\;Cos^2\;\theta}{Cos\;\theta \;Sin\;\theta}\;=\;Csc\;\theta\; Sec\;\theta$

$\displaystyle Sin^2\;\theta\;+\;Cos^2\;\theta\;=\;1$

$\displaystyle 1\;=\;1$

Did I do this right?

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Originally Posted by OzzMan

$\displaystyle Sin^2\;\theta\;+\;Cos^2\;\theta\;=\;1$

$\displaystyle 1\;=\;1$

Did I do this right?

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this is bad. when proving an identity, you cannot let the two sides interact with each other. you must leave them separate. and turn one side into the other, or bring them both to the same thing.

just change sin^2 x + cos^2 x to 1 on the left hand side and then in the very next line you can replace the left hand side with csc(x)sec(x)

Oh, you were so close!

at the point $\displaystyle \frac{sin^2(\theta)+cos^2(\theta)}{cos(\theta)sin( \theta)}$

you had the right idea, but when you replace cos^2+sin^2 with 1 you get

$\displaystyle \frac{1}{sin(\theta)cos(\theta)}$ which is $\displaystyle csc(\theta)sec(\theta)$

So when proving these. I need to pick one and only one side to mess with?

The only time we can mess with both sides is if we already know they are equivalent.

In proving identities we can't because that would be like assuming it to be true in order to show it to be true.

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Originally Posted by OzzMan

So when proving these. I need to pick one and only one side to mess with?

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I am saying, you mess with one side at a time. do not let them interact.

many problems you can mess with one side and get the other. when that's possible, do that. there are other times in which you can't get that, or you get stuck. in which case you'd leave that side, and start working on the other in order to bring the other side to something similar to where you left off the previous side

Quote:

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Originally Posted by Jhevon

I am saying, you mess with one side at a time. do not let them interact.

many problems you can mess with one side and get the other. when that's possible, do that. there are other times in which you can't get that, or you get stuck. in which case you'd leave that side, and start working on the other in order to bring the other side to something similar to where you left off the previous side

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Jhevon's right, I am sorry. You can mess with both sides, but you have to do it seperately. Like Jhevon said, they can't interact. I should have been more clear. You just can't do things like moving parts across the equal sign like adding, subtracting, multiplying or dividing both sides by something.

Thanks Jhevon!

other tips:

- you want to start messing with the more complicated side. it gives you more options to change things.

- it is USUALLY a good idea to change everything to sines and cosines. students are most familiar with those trig functions, and can therefore "see" connections easier