Prove the identity
\(\displaystyle \frac{1}{\tan x+\cot x}\equiv \sin x\cos x\).
Solution
\[
\tan x+\cot x=\frac{\sin x}{\cos x}+\frac{\cos x}{\sin x}
=\frac{\sin^2x+\cos^2x}{\sin x\cos x}
=\frac{1}{\sin x\cos x}
\Rightarrow \frac{1}{\tan x+\cot x}=\sin x\cos x.
\]
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