# Prove that (cosec A - sin A) (sec A - cos A) = 1 / tan A + cot A.

$$\text { L.H.S. }(\mathbf{cosecA}-\sin A)(\mathbf{secA}-\cos A)\\ \Rightarrow \mathbf{L . H . S}=\left(\frac{1}{\mathbf{sinA}}-\sin \mathbf{A}\right)\left(\frac{1}{\cos A}-\cos \mathbf{A}\right)\\ \Rightarrow \mathbf{L . H . S}=\left(\frac{1-\sin ^{2} \mathrm{~A}}{\sin \mathrm{A}}\right)\left(\frac{1-\cos ^{2} \mathrm{~A}}{\cos \mathrm{A}}\right)\\ \Rightarrow \mathbf{L . H . S}=\frac{\cos ^{2} \mathbf{A} \cdot \sin ^{2} \mathrm{~A}}{\mathbf{sina} \cdot \cos \mathrm{A}}\left[ \sin ^{2} \mathrm{~A}+\cos ^{2} \mathrm{~A}-1 \cos ^{2} \mathbf{A}-1-\sin ^{2} \mathrm{~A} \sin ^{2} \mathrm{~A}=1-\cos ^{2} \mathbf{A}\right]\\ \Rightarrow \mathbf{L . H . S}=\frac{\mathbf{sinA} \cdot \cos \mathrm{A}}{1}\\ \Rightarrow \mathbf{L . H} \cdot \mathbf{S}=\frac{\sin A \cdot \cos A}{\sin ^{2} A+\cos ^{2} A}\\ \Rightarrow \mathbf{L . H} \cdot \mathbf{S}=\frac{1}{\frac{\sin ^{2} A+\cos ^{2} A}{\sin A \cdot \cos A}}\\ \Rightarrow \mathbf{L . H} \cdot \mathbf{S}=\frac{1}{\frac{\sin ^{2} A}{\sin A \cos A}+\frac{\cos ^{2} A}{s \sin A \cdot \cos A}}\\ \Rightarrow \mathbf{L . H . S}=\frac{1}{\frac{\sin A}{\cos A}+\frac{\cos A}{\sin A}}\\ \Rightarrow \mathbf{L . H . S}=\frac{1}{\mathbf{tanA}+\cot \mathbf{A}}\\ =\text { R.H.S }$$

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