Question

prove that Cos A /1+sin A +1+sin A/cos A = 2 sec A

Answer 1

LHS = cosA / 1+sinA   +   1+sinA / cosA
        =(cos^2A+1+sin^2A+2sinA )  /cosA(1+sinA)
        = (2+2sinA)   /  cosA(1+sinA)
        = {2(1+sinA)}  /  cosA (1+sinA)
        = 2 / cosA
        = 2secA
RHS = 2secA
           therefore ,  LHS = RHS

Answer 2

Answer:  The proof is done below.

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

$\dfrac{\cos A}{1+\sin A}+\dfrac{1+\sin A}{\cos A}=2\sec A.$

We ill be using the following trigonometric identities:

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

We have

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

Thus, we get

$\dfrac{\cos A}{1+\sin A}+\dfrac{1+\sin A}{\cos A}=2\sec A.$

Hence proved.