Question

Tan/1-cot + cot/1-tan = 1+sec×cosec

Answer 1

Proof of Given Trignometric inequality :-

$L.H.S.\\\\\dfrac{\tan A}{1-\cot A}+\dfrac{\cot A}{1-\tan A}\\\\\\=\dfrac{\frac{\sin A}{\cos A}}{1-\frac{\cos A}{\sin A}}+\dfrac{\frac{\cos A}{\sin A}}{1-\frac{\sin A}{\cos A}}\\\\\\=\dfrac{\sin A}{\cos A}\times\dfrac{\sin A}{\sin A-\cos A}+\dfrac{\cos A}{\sin A}\times\dfrac{\cos A}{\cos A-\sin A}\\\\\\=\dfrac{\sin^2 A}{\cos A(\sin A-\cos A)}-\dfrac{\cos^2 A}{\sin A(\sin A-\cos A)}\\\\\\=\dfrac{\sin^3 A-\cos^3 A}{\sin A\cos A(\sin A-\cos A)}\\\\\\=\dfrac{(\sin A-\cos A)(\sin^2 A+\sin A\cos A+\cos^2 A)}{\sin A\cos A(\sin A-\cos A)}\\\\\\=\dfrac{\sin A\cos A+1}{\sin A\cos A}\\\\\\=1+\sec A\csc A\\\\=R.H.S.$