(cscθ−sinθ)(secθ−cosθ)=tanθ+cotθ1 PROVED.Step-by-step EXPLANATION:Consider the PROVIDED expression.(\csc\theta-\sin \theta)(\sec \theta - \cos \theta)=\dfrac{1}{\TAN \theta + \cot\theta}(cscθ−sinθ)(secθ−cosθ)=tanθ+cotθ1Consider the LHS.\begin{GATHERED}=(\csc\theta-\sin \theta)(\sec \theta - \cos \theta)\\=(\frac{1}{\sin\theta}-\sin \theta)(\frac{1}{\cos\theta} - \cos \theta)\\=(\frac{1-\sin^2\theta}{\sin\theta})(\frac{1- \cos^2 \theta}{\cos\theta})\\=(\frac{\cos^2\theta}{\sin\theta})(\frac{\sin^2 \theta}{\cos\theta})\\=\cos\theta\sin\theta\end{gathered}=(cscθ−sinθ)(secθ−cosθ)=(sinθ1−sinθ)(cosθ1−cosθ)=(sinθ1−sin2θ)(cosθ1−cos2θ)=(sinθcos2θ)(cosθsin2θ)=cosθsinθNow Consider the RHS\begin{gathered}=\dfrac{1}{\tan \theta + \cot\theta}\\=\dfrac{1}{\frac{\sin\theta}{\cos\theta} +\frac{\cos\theta}{\sin\theta}}\\\\=\dfrac{\cos\theta\sin\theta}{{\sin^2\theta}+{\cos^2\theta}}\\\\=\cos\theta\sin\theta\end{gathered}=tanθ+cotθ1=cosθsinθ+sinθcosθ1=sin2θ+cos2θcosθsinθ=cosθsinθLHS=RHSHence, proved