Loss of significance errors

Give exact ways of avoiding loss-of-significance errors in the following computations:

a. $\tan x-\tan y$, with $x\approx y$

b. $\sin x - \sin y$, with $x\approx y$

I don't know how to do a but for b I derived "$\sin(x)-\sin(y)=2\cos\frac{x+y}{2}\sin\frac{x-y}{2}$" is that correct for question b?

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For a)

$$\tan{x} - \tan{y} = \frac{\sin{(x-y)}}{\cos{x} \cos{y}} \approx \frac{\delta}{\cos^2{x}}$$

where $\delta = x-y$.

For b), you are correct.

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Thank you very much! –  Lays Feb 4 '13 at 1:21