Question

Solve it in form of cos and sin pls​

Answer 1

`\text{L.H.S}\\\\=(\cot A)/(1- \tan A) + (\tan A)/(1- \cot A)\\\\\\=\frac{\tfrac{\cos A}{\sin A}}{1- \tfrac{\sin A}{\cos A}} + \frac{\tfrac{\sin A}{\cos A}}{1-\tfrac{\cos A}{\sin A}}\\\\\\=\frac{\tfrac{\cos A}{\sin A}}{\tfrac{\cos A- \sin A}{\cos A}} + \frac{\tfrac{\sin A}{\cos A}}{\tfrac{\sin A - \cos A}{\sin A}}\\\\\\=(\cos^2 A)/(\sin A(\cos A-\sin A)) + (\sin^2 A)/(\cos A(\sin A - \cos A))\\\\\\`

`=(\cos^2 A)/(\sin A(\cos A-\sin A)) - (\sin^2 A)/(\cos A(\cos A - \cos A))\\\\\\=\frac 1{\cos A - \sin A} \left( (\cos^2 A)/(\sin A) - (\sin^2 A)/(\cos A)\right)\\\\\\=\frac 1{\cos A - \sin A} \left((\cos^3 A - \sin^3 A)/(\sin A \cos A) \right)\\\\\\=\frac 1{\cos A - \sin A} \left[((\cos A - \sin A)(\cos^2 A + \sin^2 A + \sin A \cos A))/(\sin A \cos A) \right]\\\\\\=(\cos^2 A + \sin^2 A+\sin A \cos A)/(\sin A \cos A)\\\\\\`

`=(\cos^2 A)/(\sin A \cos A)+(\sin^2 A)/(\sin A \cos A)+(\sin A \cos A)/(\sin A \cos A)\\\\\\=(\cos A)/(\sin A) + (\sin A)/(\cos A) + 1\\\\\\=\cot A + \tan A + 1\\\\\\=1 + \tan A + \cot A\\\\=\text{R.H.S}\\\\\text{Proved.}`