Hello,

I am having trouble trying to simplify the following expression;

cos^2(x) - tan^2(x) / 1 + tan^2(x)

Help is much appreciated,

Dranalion

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- Aug 6th 2009, 07:23 PMDranalionSimplification
Hello,

I am having trouble trying to simplify the following expression;

cos^2(x) - tan^2(x) / 1 + tan^2(x)

Help is much appreciated,

Dranalion - Aug 6th 2009, 07:42 PMsongoku
Hi Dranalion

You can try to apply identity : $\displaystyle 1 + \tan^2(x) = sec^2(x)$ - Aug 6th 2009, 07:58 PMDranalion
Given the identity,

**sec^2(x) = 1 + tan^2(x)**

This makes the identity:

**cos^2(x) - tan^2(x)/sec^2(x)**

Which is:

cos^2(x) - tan^2(x)/(1/cos^2(x))

How do I further simplify this expression? - Aug 6th 2009, 08:04 PMeXist
- Aug 6th 2009, 08:08 PMDranalion
Sorry, the identity is actually the divided term to be subtracted from cos^2(x), as in:

**cos^2(x) - [ tan^2(x)/(1/cos^2(x)) ]** - Aug 6th 2009, 08:14 PMeXist
Then you just have this:

$\displaystyle cos^2x - \frac{tan^2x}{sec^2x} = cos^2x - $$\displaystyle \frac{tan^2x}{\frac{1}{cos^2x}} = cos^x - (tan^2x)(cos^2x) = cos^2x - sin^2x$

-Chad - Aug 6th 2009, 08:20 PMDranalion
Thanks!

If you have**cos^2(x) - sin^2(x)**, can you use the identity:

**sin^2(x) + cos^2(x) = 1**to somehow simplify this further? Or is this as far as it can be simplified? - Aug 6th 2009, 08:23 PMeXist
Careful, we have: $\displaystyle cos^2x - sin^2x$

The identity reads: $\displaystyle cos^2x + sin^2x = 1$

Ours is not the same. In ours, we have the difference, so we cannot use that identity here. - Aug 7th 2009, 07:45 AMsongoku
Hi Dranalion

$\displaystyle cos^2x - sin^2x = \cos(2x)$