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$$1 - \frac{sin^2 A}{1 + cos A} + \frac{1 + cos A}{sin A} - \frac{sin A}{1 - cos A}$$
$$ \frac{sin^2 A}{1 + cos A} - \frac{sin A}{1 - cos A} = \frac{sin^2 A(1 - cos A)+sin A(1+cos A)}{1^{2} - cos^{2} A} =\frac{sin^2 A(1 - cos A)+sin A(1+cos A)}{ sin^{2} A} $$
$$\frac{sin^2 A(1 - cos A)+sin A(1+cos A)}{ sin^{2} A} = (1-cos A) + \frac{(1+cos A)}{sin A}$$
$$ \frac{sin^2 A}{1 + cos A} - \frac{sin A}{1 - cos A}= (1-cos A) + \frac{(1+cos A)}{sin A}$$
$$1 - \frac{sin^2 A}{1 + cos A} + \frac{1 + cos A}{sin A} - \frac{sin A}{1 - cos A}$$
$$= 1 -[(1-cos A) + \frac {1+cos A}{sin A} ] + \frac{(1+cos A)}{sin A}$$
$$= 1 -(1-cos A) - \frac {1+cos A}{sin A} + \frac{(1+cos A)}{sin A} = cos A$$
Hence Option A is the correct answer.
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