# Trig identities

1. Jul 25, 2011

### Nelo

1. The problem statement, all variables and given/known data

Prove this identity

m) Sin^2 x / sin^2 x + cos x^2 = tan^2 x / 1 + tan^ x
2. Relevant equations

http://i52.tinypic.com/105tdtk.jpg Letter (m) on the top

3. The attempt at a solution

I dont know to much about trig identities, he barely taught anything. But apparently 1 is = to cos/cos and plenty of other things.

I tried to solve the right side, the first thing i did was turn the 1 into sin^ x + cos ^ x.

Then i turned the denominator into 1-sin^x - 1 + cos^x by subbing the values of cos^x , since cos^x = 1 - sin^2 .

I know that tan^x = sin^x / cos^x , but the things that I am subbing in keep cancelling eachother out, does anyone know how to solve this?

2. Jul 25, 2011

### rock.freak667

Letter m is

$$\frac{1+tan^2x}{1-tan^2x} = \frac{1}{cos^2x-sin^2x}$$

If you take the left side and divide both the numerator and denominator by cos2x what will you get?

3. Jul 25, 2011

### Nelo

Divide the tan bracket by cos^2x?

you would get the original tan bracket, with a division of cos^2x ....

4. Jul 25, 2011

### rock.freak667

Sorry, wrong operation.

rewrite tanx as sin/cosx and then multiply the numerator and denominator by cos2x. It should easily work out.

Last edited: Jul 25, 2011
5. Jul 25, 2011

### Nelo

Its not tan x though, Its 1 + tan^2x , cant write tan^2x as sin/cos

6. Jul 25, 2011

### rock.freak667

tanx = sinx/cosx

squaring both sides

tan2x = (sinx/cosx)2 = ?

7. Jul 25, 2011

### Nelo

tru, i dont get what to do when theres a 1. it seem as tho sometimes people change the 1 into sinx^2 + cosx^2, sometimes they keep the one, honostly I dont even get how to do it. Am i always looking at both side of the equation? or just focusing on one? If i pick leftside do i ignore right side to the extent of just trying to solve for it? wth do i do with the ones?

8. Jul 25, 2011

### SammyS

Staff Emeritus
How about: taking the left side and multiplying [STRIKE]divide[/STRIKE] both the numerator and denominator by cos2x ?

9. Jul 25, 2011

### Nelo

does 1 - sin = cos?

10. Jul 25, 2011

### rock.freak667

No. But 1-sin2x=cos2x

11. Jul 26, 2011

### Nelo

f) 2sin^2x - 1 = sin^2 x - cos^2 x

I dont know how to solve this.

I tried taking the right side and simplyiing it to 1-cos^2x -1 - sin^2x ... then didnt know where to go with it,

And I have no idea where to even begin with the left side.

12. Jul 26, 2011

### Nelo

Can anyone tell me if i did this question right?

It goes :: 1 / cosx - cosx = sinx*tanx

I took the right side and simplified it to 1-cosx.

Then i took the left side and multiplied the lone cos value with cos to give me

1/cos - cos^2x / cos

1 / cos - cos^2x / cos

1-cos^2x / cos

= 1 - cos.

ls = rs

Is this correct?

13. Jul 26, 2011

### rock.freak667

If you take the left side, what can you replace '1' by? (something with sine and cosine in it)

That is one way to do it, but you should generally use one side and prove the other.

14. Jul 26, 2011

### Nelo

Are you ever supposed to recipricol subtraction into addition? Or is it jus for division into multiplication?

15. Jul 26, 2011

### rock.freak667

I am not sure what you mean by that.

When you multiply the numerator and denominator by the same quantity you are essentually multiplying by 1.

16. Jul 26, 2011

### Nelo

Is this one right?

I solved it two ways, but im uncertain if one of these ways is right or not.

f) Tan^2x - sin^2x = sin^2x*tan^2x

First:

Sin^2x
________ - 1-cos^2x = 1-cos^2x + sin^2x / cos^2x
cos^2x

^ factor out the negetive from the cos^2x
Sin^2x - 1

rs- cancel out the two cos^2x, and factor out the negitive giving -1 sin^2x in the end.

Ls = sin^2x -1 , rs = -1cos^2x

Ls = RS?

The other way is the way the teacher did it, just solving one side. etc.

Last edited: Jul 26, 2011
17. Jul 26, 2011

### rock.freak667

You could take this and then bring them both down the same denominator of cosx.

18. Jul 26, 2011

### Nelo

Like... when you have division, you take the term on the right and flip it,, do you do that for subtraction into addition as wel?

I edited my last post for a different question, please take a look @ it.

19. Jul 26, 2011

### rock.freak667

It seems like you are working with both sides at the same time. It is better to just take one side and use that side to prove the other. How you wrote it is confusing.

Sin^2x
________ - 1-cos^2x = 1-cos^2x + sin^2x / cos^2x
cos^2x

From

$$\frac{sin^2x}{cos^2x} - (1-cos^2x) = \frac{sin^2x}{cos^2x} -sin^2x$$

best to not change sin2x as yet. If you factor out the sin2x, what are you left with?

Can you see an identity that will help to get tan2x?

20. Jul 26, 2011

### Nelo

I already have that one solved, was just wondering if my way is valid to. And yes i know, my teaccher doesnt care though.