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If sum of the series \( 1+x \log _{\mid \frac{1-\sin x}{\cos x}}\left(\frac{1+\sin x}{\cos x}\right)^{1 / 2} \)
\[
+x^{2} \log _{\left|\frac{1-\sin x}{\cos x}\right|}\left(\frac{1+\sin x}{\cos x}\right)^{1 / 4}+\ldots \infty
\]
(wherever defined ) is equal to \( \frac{k(1-x)}{(2-x)} \), then \( k \) is equal to
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