The Derivative of a log of a trig function

[koudai8]

Homework Statement

I came across a question like this on a test today, and it says it would be helpful to simplify before differentiation. However, I could not find ways to simplify (1- cos(x))/(1+cos(x)).

Homework Equations

ln((1- cos (x))/(1+cos(x)))^7)

The Attempt at a Solution

I have tried to multiply by the conjugate of denominator and got (1+2cos(x)+cos(x)^2)/sin(x)^2

But I still do not see how to tackle this problem.

 

[Curious3141]

Homework Statement

I came across a question like this on a test today, and it says it would be helpful to simplify before differentiation. However, I could not find ways to simplify (1- cos(x))/(1+cos(x)).

Homework Equations

ln((1- cos (x))/(1+cos(x)))^7)

The Attempt at a Solution

I have tried to multiply by the conjugate of denominator and got (1+2cos(x)+cos(x)^2)/sin(x)^2

But I still do not see how to tackle this problem.

Hint: half-angle formula for cos x.

(EDITED: my mistake, I assumed you were required to integrate rather than differentiate!)

The half-angle formula is still a very good idea, after you apply this, apply the laws of logs to simplify further before differentiating.

 

[Premat]

Hint: half-angle formula for cos x.

Wouldn't it be easier to use the trigonometric identities for cos^2(x) and sin^2(x)?

 

[Curious3141]

Wouldn't it be easier to use the trigonometric identities for cos^2(x) and sin^2(x)?

How would those help?

 

[Premat]

How would those help?

Well, given that tan^2(u)=sin^2(u)/cos^2(u) , shouldn't it be possible to rewrite the equation into tan^2(z) , seeing as we have:

• sin^2(u) = 1/2 - 1/2 cos(2u)
  
• cos^2(u) = 1/2 + 1/2 cos(2u)

 

[Curious3141]

Well, given that tan^2(u)=sin^2(u)/cos^2(u) , shouldn't it be possible to rewrite the equation into tan^2(z) , seeing as we have:

• sin^2(x) = 1/2 - 1/2 cos(2x)
  
• cos^2(x) = 1/2 + 1/2 cos(2x)

What's "z"?

The half angle formulae are basically equivalent to the double angle formulae, and they allow an immediate simplification.

Although, frankly, if differentiation is what's required, I don't think prior simplification helps a whole lot.

 

[Premat]

What's "z"?

The half angle formulae are basically equivalent to the double angle formulae, and they allow an immediate simplification.

Although, frankly, if differentiation is what's required, I don't think prior simplification helps a whole lot.

z $\in$ ℝ , didn't want it to get mixed up with x

What does it simplify to using the half-angle formulae? Using the previously stated identities it should be possible to simplify the expression inside the logarithm to tan(x/2)

EDIT: That's tan^2(x), not tan(x)!

 

[Curious3141]

z $\in$ ℝ , didn't want it to get mixed up with x

What does it simplify to using the half-angle formulae? Using the previously stated identities it should be possible to simplify the expression inside the logarithm to tan(x/2)

Should that be $\displaystyle \tan^2 \frac{x}{2}$? Because that's what I get with the half-angle formula.

Further simplification with the laws of logs is trivial.

 

[Premat]

Should that be $\displaystyle \tan^2 \frac{x}{2}$? Because that's what I get with the half-angle formula.

Further simplification with the laws of logs is trivial.

Yeah, you're right, forgot to square it in my post. I'll edit it.