Verifying Trig Identity: csc(x) + sec(x) = cot(x) + tan(x)

[Trizz]

Homework Statement

Verify the identity:

csc(x) + sec(x) = cot(x) + tan(x)
sin(x) + cos(x)

Homework Equations

sin^2 + cos^2 = 1
tan^2 + 1 = sec^2
1 + cot^2 = csc^2

The Attempt at a Solution

First i changed everything to sin and cos to try and make it more clear. Let me know if I should bypass this:

1/sin(x) + 1/cos(x) = cot(x) + tan(x)
sin(x) + cos(x)

I'm not really sure where to go from here...should i try to make a common denominator?

 

[rock.freak667]

Multiply the numerator and denominator by sin(x)cos(x)

 

[Trizz]

Thanks rock.freak!

Ok i did that and got:

cos(x) + sin(x)
sin(x)^2 + cos(x)^2


does that look right?

 

[Trizz]

Or is it this?

sin(x) + cos(x)
sin(x)^2cos(x) + sin(x)cos(x)^2

 

[Mentallic]

Or is it this?

sin(x) + cos(x)
sin(x)^2cos(x) + sin(x)cos(x)^2

It's this one. The other is obviously not true since sin^2x+cos^2x=1 so the denominator would become 1.

Don't expand out the denominator, instead, see if you can cancel any common factors.
After this step, it should be easier to work on the right-hand side to try get the left-hand side.

 

[Trizz]

Thanks Mentallic.

So I kind of looked at the denominator like this (a^2*b)(a*b^2).

So I should be able to pull out one "a" and one "b", right?

Does this seem right?

sin(x) + cos(x)
sin(x)cos(x)(sin(x)^2 + cos(x)^2)

 

[Mentallic]

Actually the denominator is more like a^2b+b^2a.

Now you can factor out ab to get ab(a+b). Do you see how this works?

For the denominator it should be sinx.cosx\left(sinx+cosx\right) so do you see what can be canceled in that fraction?

 

[Trizz]

Thanks a lot Mentallic.

Yes, I see what you did on the factoring. I did that too quickly.

So with the new denominator, it would give me:

sin(x) + cos(x)
sin(x).cos(x) (sin(x) +cos(x))


It seems like you can cancel out the sin(x) + cos(x) of the numerator with the same thing in parenthesis in the denominator, but am I wrong? Can't you only cancel out things that are multiplied or divided, not attached through addition?

If so, I'm lost on what to cancel out.

 

[Bohrok]

It looks like there was no need to distribute sinxcosx in the denominator at the beginning; just better to have left it like (sinx + cosx)sinxcosx.

Yes, sinx + cosx will cancel, then you have 1/(sinx*cosx). Then use the identity 1 = sin²x + cos²x for the numerator.

 

[Trizz]

Thanks Bohrok.

I'm still a little confused about how that would verify the identity

Are you saying to set up the fraction like this now?

sin^2x + cos^2x
sinx*cosx

I see the exponent and denominator would cancel out, but how would that give me cot(x) + tan(x) as the final solutions

 

[Mentallic]

Thanks a lot Mentallic.

Yes, I see what you did on the factoring. I did that too quickly.

So with the new denominator, it would give me:

sin(x) + cos(x)
sin(x).cos(x) (sin(x) +cos(x))


It seems like you can cancel out the sin(x) + cos(x) of the numerator with the same thing in parenthesis in the denominator, but am I wrong? Can't you only cancel out things that are multiplied or divided, not attached through addition?

If so, I'm lost on what to cancel out.

Yes you're allowed to cancel the sinx+cosx from the numerator and denominator. They're both the same factor.

e.g. if we have a fraction such as:

\frac{xy}{x}
we can cancel x from both the numerator and denominator because they're common factors.
This rules works for whatever x may stand for, such as something more complicated like sina+cosb
Then the fraction would be: \frac{(sina+cosb)y}{sina+cosb} and now you can cancel in the same fashion.


Ok so after you cancel the sinx+cosx you will have \frac{1}{sinx.cosx}.
It is a little hard to see what you could do with this to try get the RHS, so it would be easier to work on the RHS from this point.

 

[Bohrok]

Are you saying to set up the fraction like this now?

sin^2x + cos^2x
sinx*cosx

I see the exponent and denominator would cancel out, but how would that give me cot(x) + tan(x) as the final solutions

Then simplify the fraction using the fact that (a + b)/c = a/c + b/c. You could work on the right hand side as Mentallic said so you can see how it should work out.

 

[Trizz]

Thank you guys for being so kind and giving me your time! I think I can get the rest from here.

 

[Mentallic]

No problem, good luck!