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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{We} \mathrm{have} \\ \mathrm{L}.\mathrm{H}.\mathrm{S}.=\mathrm{cosx}+\sin \left(\frac{3\mathrm{\pi }}{2}+\mathrm{x}\right)-\sin \left(\frac{3\mathrm{\pi }}{2}-\mathrm{x}\right)+\cos \left(\mathrm{\pi }+\mathrm{x}\right) \\ \mathrm{As} \sin \left(\frac{3\mathrm{\pi }}{2}+\mathrm{x}\right)=\sin \left(\frac{3\mathrm{\pi }}{2}-\mathrm{x}\right)=-\mathrm{cosx} \mathrm{and} \cos \left(\mathrm{\pi }+\mathrm{x}\right)=-\mathrm{cosx}, \mathrm{hence} \\ \mathrm{we} \mathrm{get} \\ \mathrm{L}.\mathrm{H}.\mathrm{S}.=\mathrm{cosx}-\mathrm{cosx}-\left(-\mathrm{cosx}\right)-\mathrm{cosx} \\ =\mathrm{cosx}-\mathrm{cosx}+\mathrm{cosx}-\mathrm{cosx} \\ =0 =\mathrm{R}.\mathrm{H}.\mathrm{S}. \left[\mathrm{Proved}\right] \end{gathered}$$

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