1. ## Simplification

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

2. Hi Dranalion

You can try to apply identity : $\displaystyle 1 + \tan^2(x) = sec^2(x)$

3. 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?

4. Originally Posted by songoku
Hi Dranalion

You can try to apply identity : $\displaystyle 1 + \tan^2(x) = sec^2(x)$

After you apply this you have:

$\displaystyle \frac{cos^2(x) - tan^2(x)}{sec^2(x)} = \frac{cos^2(x) - tan^2(x)}{\frac{1}{cos^2(x)}} = (cos^2x - tan^2x)(cos^2x) = cos^4x - sin^2x$

Hope this helps

5. 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)) ]

6. 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$

7. 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?

8. 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.

9. Hi Dranalion

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