Question

cot/cosec+1 + cosec+1/cot =2sec

Answer 1

Hence proved hope its helpful

Answer 2

Answer:

Given,

$\dfrac{\cot x}{\cosec x+1}+\dfrac{\cosec x+1}{\cot x}=2\sec x$

We proceed by taking the left hand side:

$L.H.S.=\dfrac{\cot x}{cosec \;x+1}+\dfrac{cosec \;x+1}{\cot x}\\L.H.S=\dfrac{\frac{\cos x}{\sin x}}{\frac{1}{\sin x}+1}+\dfrac{\frac{1}{\sin x}+1}{\frac{\cos x}{\sin x}}\\L.H.S=\dfrac{\frac{\cos x}{\sin x}}{\frac{1+\sin x}{\sin x}}+\dfrac{\frac{1+\sin x}{\sin x}}{\frac{\cos x}{\sin x}}\\L.H.S=\dfrac{\cos x}{1+\sin x}+\dfrac{1+\sin x}{\cos x}\\L.H.S=\dfrac{cos^2x+(1+\sin x)^2}{(1+\sin x)\cos x}\\L.H.S=\dfrac{(\cos^2x+\sin^2x)+1+2\sin x}{(1+\sin x)\cos x}\\L.H.S=\dfrac{2+2\sin x}{(1+\sin x)\cos x}\\L.H.S=\dfrac{2(1+\sin x)}{(1+\sin x)\cos x}\\L.H.S=\dfrac{2}{\cos x}\\L.H.S=2\sec x\\L.H.S=R.H.S$