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**HD09** How can I calculate it for $\displaystyle \frac{1}{1+cos^2(x)}$ by using the fact that $\displaystyle \frac{1}{1+x^2} = 1 - x^2 + x^4 - ...$?

I tried letting u = cos(x), then

$\displaystyle \frac{1}{1+cos^2(x)} = \frac{1}{1+u^2} = 1 - u^2 + u^4 - ... = 1 - cos^2(x) + cos^4(x) - ...$

But I don't think this is right because the first term should be 0.5, not 1... and I don't see how a -0.5 might pop out of this series of cos terms....and even if it somehow does, I think this question is not meant to be that difficult...

Any ideas?

Thanks