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# Trigonometric Identities Help!!

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1. So I started AS maths last week and I'm having some trouble with identities. Please please please can someone try to help me!

1. (1/sinx) - tanxcosx = cos^2x/sinx
2. tanxsinx/1-cosx = 1 + 1/cosx
3. (cosx+sinx)^2 + (cosx-sinx)^2 =2
4. sinxtanx + cosx = 1/cosx
5. 1/sinx - 1/tanx = 1 - cosx/sinx

Any help would be greatly appreciated!
2. (Original post by Alex.trin)
So I started AS maths last week and I'm having some trouble with identities. Please please please can someone try to help me!

1. (1/sinx) - tanxcosx = cos^2x/sinx
2. tanxsinx/1-cosx = 1 + 1/cosx
3. (cosx+sinx)^2 + (cosx-sinx)^2 =2
4. sinxtanx + cosx = 1/cosx
5. 1/sinx - 1/tanx = 1 - cosx/sinx

Any help would be greatly appreciated!
What are you stuck on exactly? At this point you should know that:

and work from these.
3. (Original post by RDKGames)
What are you stuck on exactly? At this point you should know that:

and work from these.
Yes I found those in the textbook but I don't understand how to rearrange them to simplify those equations
4. (Original post by Alex.trin)
Yes I found those in the textbook but I don't understand how to rearrange them to simplify those equations
I'll do the first one for you as an example:

First rewrite as . You can see that multiplying this by cancels out the denominator right away.

You are left with:

Now multiply through by sin(x) to cancel the 1/sinx and you are left with: which is a true identity.

Do the similar procedure for the other ones.
5. (Original post by RDKGames)
I'll do the first one for you as an example:

First rewrite as . You can see that multiplying this by cancels out the denominator right away.

You are left with:

Now multiply through by sin(x) to cancel the 1/sinx and you are left with: which is a true identity.

Do the similar procedure for the other ones.
Oh my goodness you are a genius! I'll have a go at the others and see how I get on. Thank you!!
6. (Original post by RDKGames)
I'll do the first one for you as an example:

First rewrite as . You can see that multiplying this by cancels out the denominator right away.

You are left with:

Now multiply through by sin(x) to cancel the 1/sinx and you are left with: which is a true identity.

Do the similar procedure for the other ones.
It is very bad practice to assume that an identity is true and manipulate it to arrive at something true.

If you do this you have to be sure that every step you make is reversible which isn't always obvious.

With trig identities you should always start from one side and show that the other side is true.
7. Right I'm stuck again. On number 3, cos^2x + sin^2 simplifies to 1, but how do I simplify cos^2x - sin^2x?
8. (Original post by Alex.trin)
Oh my goodness you are a genius! I'll have a go at the others and see how I get on. Thank you!!
Please do not do what RDKGames has done for trig identities.

You should always start from one side of a trig identity and aim to arrive at the other side. Never manipulate the whole identity like you would an equation.
9. (Original post by notnek)
It is very bad practice to assume that an identity is true and manipulate it to arrive at something true.

If you do this you have to be sure that every step you make is reversible which isn't always obvious.

With trig identities you should always start from one side and show that the other side is true.
The question didn't really make it clear and I assumed they want you to prove whether something turns out to be a true identity or not.
10. (Original post by Alex.trin)
Right I'm stuck again. On number 3, cos^2x + sin^2 simplifies to 1, but how do I simplify cos^2x - sin^2x?
Show your working, it shouldn't simplify down to a minus. Also as the person above said, leave the right hand side along and work with left one.
11. (Original post by notnek)
Please do not do what RDKGames has done for trig identities.

You should always start from one side of a trig identity and aim to arrive at the other side. Never manipulate the whole identity like you would an equation.
Ok, please may you show me how you'd answer the question then?
12. (Original post by RDKGames)
The question didn't really make it clear and I assumed they want you to prove whether something turns out to be a true identity or not.
That doesn't make sense.

I'm busy now but if you have time can you please show the OP how to do question 1 using the correct method?
13. (Original post by RDKGames)
Show your working, it shouldn't simplify down to a minus. Also as the person above said, leave the right hand side along and work with left one.
Oh yes I see now. (-sin^2x)^2 is positive, not negative. Thanks again
14. (Original post by notnek)
That doesn't make sense.

I'm busy now but if you have time can you please show the OP how to do question 1 using the correct method?
To me it does, but okay.

(Original post by Alex.trin)
Oh yes I see now. (-sin^2x)^2 is positive, not negative. Thanks again
Q1 corrected:

Rewrite tanx as sinx over cosx, the cosx cancels with the denominator as before. The RHS wants the answer to be as something over sinx, so you need to combine the LHS into a single fraction under the same denominator. The first term already has sinx as denominator, we can leave it. The second term, sinx, can be turned into a fraction with denominator sinx by:

Now add the two fractions and you get: at which point we need to show that the numerator is equal to and indeed this is true from rearranging
15. (Original post by RDKGames)
To me it does, but okay.

Q1 corrected:

Rewrite tanx as sinx over cosx, the cosx cancels with the denominator as before. The RHS wants the answer to be as something over sinx, so you need to combine the LHS into a single fraction under the same denominator. The first term already has sinx as denominator, we can leave it. The second term, sinx, can be turned into a fraction with denominator sinx by:

Now add the two fractions and you get: at which point we need to show that the numerator is equal to and indeed this is true from rearranging
Oh right I see now! So would the answer be cos^2x/sinx = cos^2x/sinx??
16. (Original post by Alex.trin)
Oh right I see now! So would the answer be cos^2x/sinx = cos^2x/sinx??
Exactly. With these what you are essentially doing is turning the left hand side of the identity into the right hand side.
17. (Original post by RDKGames)
To me it does, but okay.
Call the identity a statement and your (true) hypothesis (that's stuff like sin^2 + cos^2 = 1, etc...)

You want to show that . You're trying to show . These are the same things only when is true.

For a concrete example, take to be the statement and the statement .

Then whilst it's true that it does not hold that .
18. (Original post by RDKGames)
Exactly. With these what you are essentially doing is turning the left hand side of the identity into the right hand side.
So I've moved on to question 4 now. I've turned sinx into tanx/cosx and tanx into sinx/cosx. Then I've combined the two fractions to make tanxsinx/cosx. Now I'm stuck because I can't just cross out the cosx's. Please please please can someone help!
19. (Original post by Alex.trin)
So I've moved on to question 4 now. I've turned sinx into tanx/cosx and tanx into sinx/cosx. Then I've combined the two fractions to make tanxsinx/cosx. Now I'm stuck because I can't just cross out the cosx's. Please please please can someone help!
Er, just turn tan(x) into sinx over cosx. Don't try to go from sin or cos into tan right away, deal with the basic functions first. The first term should then read sin squared over cos which is okay as far as the RHS is concerned, then deal with the second term and get it under the same denominator.
20. (Original post by Alex.trin)
So I've moved on to question 4 now. I've turned sinx into tanx/cosx and tanx into sinx/cosx. Then I've combined the two fractions to make tanxsinx/cosx. Now I'm stuck because I can't just cross out the cosx's. Please please please can someone help!
Why would you want to do that? Think about it! You're starting from the left hand side which is in terms of tans, sines and cosines and you wnat to show that's the same thing as the right hand side which is made up only of cosines. So the logical thing to do in the RHS is to turn the tan into a sine and cosine, and then hopefully get rid of sines altogether.

So try: - now do common denominators, then add the numerator and you get something nice in the numerator that simplies very nicely and gets you what you want!

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