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If $ \int e^x \bigg(\large\frac{1- \sin x}{1-\cos x}\bigg)$$dx-f(x)+constant$ then $f(x)=$

\[\begin {array} {1 1} (1)\;e^x \cot \bigg(\frac{x}{2} \bigg) & \quad (2)\;e^{-x} \cot \bigg(\frac{x}{2} \bigg) \\ (3)\;-e^x \cot \bigg(\frac{x}{2} \bigg) & \quad (4)\;-e^{-x} \cot \bigg(\frac{x}{2} \bigg) \end {array}\]

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