Question

{sinA/1+cosA + 1+cosA/sinA}.{cosA/1+sinA+1+sinA/cosA}= 4secA.cosecA

Answer 1

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Answer 2

Answer:  Proved.

Step-by-step explanation:  We are given to prove the following:

$\left(\dfrac{\sin A}{1+\cos A}+\dfrac{1+\cos A}{\sin A}\right)\left(\dfrac{\cos A}{1+\sin A}+\dfrac{1+\sin A}{\cos A}\right)=4\sec A\csc A.$

We will need the following trigonometric identities in the proof:

$(i)~\cos^2A+\sin^2A=1,\\\\(ii)~\dfrac{1}{\cos A}=\sec A,\\\\(iii)~\dfrac{1}{\sin A}=\csc A.$

We have

$L.H.S.\\\\=\left(\dfrac{\sin A}{1+\cos A}+\dfrac{1+\cos A}{\sin A}\right)\left(\dfrac{\cos A}{1+\sin A}+\dfrac{1+\sin A}{\cos A}\right)\\\\\\=\left(\dfrac{\sin^2A+(1+\cos^2A)}{(1+\cos A)\sin A}\right)\left(\dfrac{\cos^2A+(1+\sin^2A)}{(1+\sin A)\cos A}\right)\\\\\\=\dfrac{(\sin^2A+1+2\cos A+\cos^2A)(\cos^2A+1+2\sin A+\sin^2A)}{(1+\cos A)(1+\sin A)\cos A\sin A}\\\\\\=\dfrac{(2+2\cos A)(2+2\sin A)}{(1+\cos A)(1+\sin A)\cos A\sin A}\\\\\\=\dfrac{4(1+\cos A)(1+\sin A)}{(1+\cos A)(1+\sin A)\cos A\sin A}\\\\\\=\dfrac{4}{\cos A\sin A}\\\\=4\sec A\csc A\\\\=R.H.S.$

Thus, the given equality is proved.