Proving trigonometric identities

[DJ-Smiles]

Homework Statement

Prove that:

(1-tanθ)/(1+tanθ)=(cotθ-1)/(cotθ+1)

Homework Equations

Trig Identities:

tanθ= sinθ/cosθ
cotθ= cosθ/sinθ
1+tanθ=secθ
1+cotθ=cosecθ

The Attempt at a Solution

These sorts of equations are coming up a lot and I am having trouble understanding what I have to do exactly, I have seen people cross multiply which cannot really work considering we haven't got proof that they equal each other and hence can't cross multiply. I have also attempted changing 1+tanθ to secθ and 1+cotθ to cosecθ but I am having no luck. I know there is a certain way to do it, I am just unsure of what this way is.

If possible could someone give me a guide on how to do it rather than just hint at things? I am coming up to exams in about two weeks and I don't have time to muck around, I need to make sure that I know everything that could be on the test.

Thanks in advance for your inputs!

 

[Mark44]

Homework Statement

Prove that:

(1-tanθ)/(1+tanθ)=(cotθ-1)/(cotθ+1)

Homework Equations

Trig Identities:

tanθ= sinθ/cosθ
cotθ= cosθ/sinθ

The two below aren't identities.

1+tanθ=secθ
1+cotθ=cosecθ

The Attempt at a Solution

These sorts of equations are coming up a lot and I am having trouble understanding what I have to do exactly, I have seen people cross multiply which cannot really work considering we haven't got proof that they equal each other and hence can't cross multiply. I have also attempted changing 1+tanθ to secθ and 1+cotθ to cosecθ but I am having no luck.

Which is to be expected, because the actual identies are
1+tan²θ = sec²θ and

1+cot²θ = csc²θ

I know there is a certain way to do it, I am just unsure of what this way is.

If possible could someone give me a guide on how to do it rather than just hint at things? I am coming up to exams in about two weeks and I don't have time to muck around, I need to make sure that I know everything that could be on the test.

Thanks in advance for your inputs!

Start on one side (usually the side that seems most complicated, but that's subjective), and use identities to arrive at what you have on the other side.

For your problem, one approach would be to write all of the tan and cot functions in terms of sin and cos, and go from there.

 

[DJ-Smiles]

Yeah sorry, I knew that but I was just rushing to write this down. Ok I will try that. Now say a different situation comes up say it was : (1-sinx)/(1+sinx)=(1-cosx)/(1+cos). Not sure if that is doable but something along those lines like instead of cot and tan it was cos or sin? what would i do then?

 

[Mark44]

Yeah sorry, I knew that but I was just rushing to write this down. Ok I will try that. Now say a different situation comes up say it was : (1-sinx)/(1+sinx)=(1-cosx)/(1+cos). Not sure if that is doable but something along those lines like instead of cot and tan it was cos or sin? what would i do then?

First off, you can't just make up something and try to show it's an identity. In this case, your equation is not an identity. To see that, note that if x = 0, the left side value is 1, and the right side value is 0.

 

[Mark44]

Moving thread to the Precalc section...

 

[Saitama]

For your problem, one approach would be to write all of the tan and cot functions in terms of sin and cos, and go from there.

Wouldn't that be much easier if you write tan as 1/cot?

 

[Millennial]

Wouldn't that be much easier if you write tan as 1/cot?

Or write cot as 1/tan. Yes, I think so.