Prove the following trigonometrical identities: \[ \frac{\tan x}{1-\cot x}+\frac{\cot x}{1-\tan ...

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Prove the following trigonometrical identities:
\[
\frac{\tan x}{1-\cot x}+\frac{\cot x}{1-\tan x}=1+\tan x+\cot x=1+\sec x \operatorname{cosec} x
\]
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