We know that cotx repeats same value after an interval 2π
L.H.S.= \( cos(\dfrac{3\pi}{2}+x)(Cos 2π+x)\)
\( [cot(\dfrac{3\pi}{2}-x)+(Cot 2π+x)]=1\)
=sinxcosx[tanx+cotx]
Substituting \( tanx=\dfrac{sinx}{cosx}\) and
\(cotx=\dfrac{cosx}{sinx}\),
L.H.S=sinxcosx \((\dfrac{sinx}{cosx}+\dfrac{cosx}{sinx})\)
\((sinxcosx) [\dfrac{sin^2x+cos^2x}{sinxcosx}]\)
= 1
= R.H.S.
Hence proved.
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