If \( \int_{\text {then }} \sqrt{\operatorname{cosec} x+1} d x=k f ...

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If \( \int_{\text {then }} \sqrt{\operatorname{cosec} x+1} d x=k f o g(x)+c \), where \( k \) is a real constant,
\( \mathrm{P} \)
(1) \( k=-2, f(x)=\cot ^{-1} x, g(x)=\sqrt{\operatorname{cosec} x-1} \)
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(2) \( k=-2, f(x)=\tan ^{-1} x, g(x)=\sqrt{\operatorname{cosec} x-1} \)
(3) \( k=2, f(x)=\tan ^{-1} x, g(x)=\frac{\cot x}{\sqrt{\operatorname{cosec} x-1}} \)
(4) \( k=2, f(x)=\cot ^{-1} x, g(x)=\frac{\cot x}{\sqrt{\operatorname{cosec} x+1}} \)
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