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I would like to know the clarification of the following.

If the follwoing one is correct, we can find cosix and sin ix series very quickly.

We know that, Cosx + i sinx = e^ix

If we replace x by ix on both sides, we get; cos ix + i sin i x = 1/e^x

we know that e^-x series. In this by taking real part and imaginary part and then equating to LHS, can we get cos ix and sinix series. If it is wrong? why it is wrong...justify. Thanks in advance

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    It's wrong because $\cos ix$ and $\sin ix$ are **not** strictly real, so they won't correspond to the real/imaginary parts of the right side. (The RHS doesn't even *have* an imaginary part for real $x$.)2011-10-20
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    It's not clear to me what the point of inserting $\mathrm i$ is here. The Euler formula itself is a good basis for relating the series for the exponential and sine and cosine; why this transformation attempt?2011-10-20
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    (Well, technically $\cos ix$ is always real, but $\sin ix$ is purely imaginary.)2011-10-20

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