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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Hi Dranalion You can try to apply identity : $\displaystyle 1 + \tan^2(x) = sec^2(x)$
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?
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 -Chad
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)) ]
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
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?
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.
Hi Dranalion $\displaystyle cos^2x - sin^2x = \cos(2x)$
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