Prove the identity
\((1+\sec x)(\csc x-\cot x) \equiv \tan x\).
Solution
\[
\csc x-\cot x=\frac{1-\cos x}{\sin x},\qquad
1+\sec x=1+\frac{1}{\cos x}=\frac{1+\cos x}{\cos x}.
\]
Thus
\[
(1+\sec x)(\csc x-\cot x)=
\frac{(1-\cos x)(1+\cos x)}{\sin x\cos x}
=\frac{1-\cos^2x}{\sin x\cos x}
=\frac{\sin^2x}{\sin x\cos x}=\tan x.
\]
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