different answers for derivative?

**dhtikna** · March 1st 2013, 05:24 AM

Take this trigonometric function:

$\frac{d \(sin{(\cos{\theta})})^2}{d \theta}}$

Using the chain rule i think it becomes:

$-2\theta\sin{(\cos{\theta})}\sin{\theta}$

But if we use the identity:

$\sin^2{\theta} = 1-\cos^2{\theta}$

then the equation becomes

$\frac{d (1-\theta^2)}{d \theta}} = -2\theta$

But if,

$-2\theta = -2\theta\sin{(\cos{\theta})}\sin{\theta}$

then,

$\sin{(\cos{\theta})}\sin{\theta} = 1$

Which doesn't appear so. Why am I getting different answers for the same derivative? Did I do something wrong in the above steps?

I accidentally took :

$\cos(\cos\theta) = \theta$

P.S. This is my first post and using [tex] is VERY slow and irritating, is there a faster way to insert math expressions

**Plato** · March 1st 2013, 05:46 AM

Originally Posted by dhtikna

Take this trigonometric function:
$\frac{d \(sin{(\cos{\theta})})^2}{d \theta}}$

Using the chain rule i think it becomes:
$-2\theta\sin{(\cos{\theta})}\sin{\theta}$

This part of the post is correct. But none of the rest makes any sense.

It is the case that $\sin^2(\cos(\theta))=1-\cos^2(\cos(\theta))$ . But that cannot be simplified.

They both have the same derivative.

**HallsofIvy** · March 1st 2013, 05:58 AM

$cos(arccos(x))= x$ but you have $cos(cos(\theta))$ which is NOT $\theta$ .

**dhtikna** · March 1st 2013, 06:38 AM

I Can't belive i was so stupid. I hadn't been doing trig for a long time so i messed up when I took:

$\cos({\cos\theta}) = \theta$

By the way, any have a answer to my P.S.:

P.S. This is my first post and using [tex] is VERY slow and irritating, is there a faster way to insert math expressions

**Prove It** · March 1st 2013, 03:07 PM

Originally Posted by dhtikna

I Can't belive i was so stupid. I hadn't been doing trig for a long time so i messed up when I took:

$\cos({\cos\theta}) = \theta$

By the way, any have a answer to my P.S.:

P.S. This is my first post and using [tex] is VERY slow and irritating, is there a faster way to insert math expressions

Yes, get good at copying and pasting the preamble, learn the code and type faster :P

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