Trig identities: does this equal zero?

• Never.ever
In summary, the student is trying to solve a calculus problem, but is stuck and needs to find an earlier mistake.

Homework Statement

This is actually many steps through a calculus problem involving trig functions. I have not included the problem because I'm trying very hard to figure it out on my own (at least as far as it's possible). I've found the answer I'm looking for, but it's attached to a bunch of random junk. I'll be fine as long as the following identity is true:

2(tanx(sinx)^2-cotx(cosx)^2) + 2(tanx+cotx) + (tanx)^2 + (cotx)^2 + 2 = 0

Problem is, I just can't seem to make it work. I've triple-checked all the steps up to this point, so I'm pretty confident that the numbers here are correct. Is there any way that the above identity is true?

Homework Equations

(cosx)^2+(sinx)^2 = 1
(tanx)^2 + 1 = (secx)^2
(cotx)^2 + 1 = (cscx)^2

The Attempt at a Solution

Working piecemeal, trying to find hidden identities within the above equation. For instance:
dividing 2sinxcosx from (2cotx-cotx(cosx)^2) produces 2sinxcosx((cscx)^2-(cotx)^2=2sinxcosx(1)=sin2x
I can also find both (1+(tanx)^2) and (1+cotx)^2) in the equation, producing (secx)^2 and (cscx)^2. However, none of these seem to be steps in the right direction.

It seems like what I need is to find some way to pull -2 out of an identity that equals 1, which would allow me to get rid of that 2 on the end. Then I need to arrange the rest of the identities so that they equal zero. I'm beginning to suspect this isn't possible, but my brain just can't let go.

Nevermind. I started the problem over, taking the derivative using quotient rule rather than product rule and discovered that I had two signs flipped around. This is exactly the problem I kept slamming up against with the above, so I'm hoping this will solve everything. If'n it does, let me just say I really appreciate the service you guys have here, keep up the great work!

So I've redone the entire equation, and have now become stuck at a different point. Here's what I have left:

[ ((cotx)^2 - (tanx)^2) + 8(cosxsinx) +8(sinx)^2]/(1+tanx)^2(1+cotx)^2= -cos2x

I'm pretty sure that denominator just equals 4, so what I'm looking for is the numerator to equal -4((cosx)^2-(sinx)^2) and then I've got my answer. But I can't find that in here anywhere. Am I missing something?

Well, I'm really losing hope now. Of course, I found the denominator does not equal zero, but is in fact just another giant mess. I simplified the numerator by dividing (sinx)^2 and (cotx)^2 by (sinx)^2 to get (sinx)^2*(1+(secx)^2)=(sinx)^2(tanx)^2, then divided that and my original (tanx)^2 by -(tanx)^2 to get -tanx(1-(sinx)^2) = -(tanx)^2(cosx)^2 = -(sinx)^2. So after that, the numerator becomes:

8(cosxsinx) + 6(sinx)^2

And looks even more impenetrable than before. Ugh.

Never.ever said:
So I've redone the entire equation, and have now become stuck at a different point. Here's what I have left:

[ ((cotx)^2 - (tanx)^2) + 8(cosxsinx) +8(sinx)^2]/(1+tanx)^2(1+cotx)^2= -cos2x

I'm pretty sure that denominator just equals 4, so what I'm looking for is the numerator to equal -4((cosx)^2-(sinx)^2) and then I've got my answer. But I can't find that in here anywhere. Am I missing something?

If I substitute x = pi/4, I get 1/2 = 0, so I think you made an earlier mistake.

willem2 said:
If I substitute x = pi/4, I get 1/2 = 0, so I think you made an earlier mistake.

Doh!

Thank you. For some reason, such an obvious test never occurred to me :)

1. What are trig identities?

Trig identities are mathematical equations that involve trigonometric functions, such as sine, cosine, and tangent. They are used to simplify and manipulate trigonometric expressions.

2. Why do we use trig identities?

We use trig identities to solve trigonometric equations, simplify expressions, and prove mathematical theorems. They allow us to transform complex trigonometric expressions into simpler forms, making them easier to work with.

3. How do I know which trig identity to use?

Knowing which trig identity to use depends on the specific problem you are trying to solve. It is important to have a good understanding of the properties of trigonometric functions and be able to recognize patterns in the given expression to determine which identity would be most useful.

4. Can I use trig identities to solve for any variable?

Yes, trig identities can be used to solve for any variable in a trigonometric expression. They can be used to manipulate equations involving sine, cosine, tangent, and other trigonometric functions to solve for a specific variable.

5. How do I check if a trig identity equals zero?

To check if a trig identity equals zero, you can substitute the given values into the expression and simplify it. If the simplified expression results in 0, then the identity is true. You can also use basic trigonometric identities, such as the Pythagorean identity, to transform the expression and see if it equals zero.