Simplifying an incredibly annoying equation

[Titandwedebil]

Homework Statement

Simplify: 1 + (cot²x) - (cos²x) - (cos²x)(cot²x)

Homework Equations

All reciprocal, quotient, and Pythagorean identities.

The Attempt at a Solution

I have spent SO much time trying to figure this out, the answer is "1" but I cannot get it! I managed to get "1 + (cot²x) - (cos²x)" equal to one but the other side is impossible! :(

 

[dextercioby]

Re-write as

1-cos^2 x + cot^2 x - cos^2 x cot^2 x = sin^2 x + cot^2 x (1 - cos^2 x) = sin^2 x + ...

 

[WannabeNewton]

Represent all of the trig functions in terms of sin and csc based on identities. Do you remember what cot^2(x) is equal to?

 

[Titandwedebil]

@WannabeNewton = Cot²x is Cosx²/Sinx²

I'm just confused how you guys knew that stuff right off the bat. The way that I worked up to this point was simply eliminating anything to do with tangent and going from there. How do you guys know that you're supposed to work with csc and sin as you both said?

 

[WannabeNewton]

Use cot^2(x) = csc^2(x) - 1. Sometimes its helpful to just get all of the trig functions in terms of one trig function and seeing what cancels out if you want just a constant as a result.

 

[Mark44]

I'm just confused how you guys knew that stuff right off the bat.

What you cavalierly put as "All reciprocal, quotient, and Pythagorean identities." under Relevant equations, some of us know by heart.

In this case,
sin(x) = sin(x)
cos(x) = cos(x)
tan(x) = sin(x)/cos(x)

csc(x) = 1/sin(x)
sec(x) = 1/cos(x)
cot(x) = cos(x)/sin(x)

and
sin²(x) + cos²(x) = 1
tan²(x) + 1 = sec²(x)
1 + cot²(x) = csc²(x)

 

[Titandwedebil]

@Mark - Oh! No, I knew that most of you would know the basic identities, I was just questioning the methods that they used to get started.

I've gone through it again, and I'm still coming up short.

"Re-write as

1-cos^2 x + cot^2 x - cos^2 x cot^2 x = sin^2 x + cot^2 x (1 - cos^2 x) = sin^2 x + ..."

I turned that cot into cos/sin and tried multiplying it with "(1-cos^2 x)", which ended up not working...

This is where I ended up at...
1-(cos² x) + (cos²x)-(cos⁴x)/(cos²x) - 1

These simplification problems are making me rip my hair out!

 

[eumyang]

I've gone through it again, and I'm still coming up short.

"Re-write as

1-cos^2 x + cot^2 x - cos^2 x cot^2 x = sin^2 x + cot^2 x (1 - cos^2 x) = sin^2 x + ..."

I turned that cot into cos/sin and tried multiplying it with "(1-cos^2 x)",

No, don't do that.
$$1 - \cos^2 x$$
simplifies to something.

Note that not only should you know the three Pythagorean identities, but also their variations. For instance,
$$1 + \tan^2 x = \sec^2 x$$
, but if you subtract 1 from both sides,
$$\tan^2 x = \sec^2 x - 1$$
. So if in another problem you encounter the expression "sec²x - 1," you can replace it with "tan²x."

Now figure what
$$1 - \cos^2 x$$
simplifies to.