Question

$\frac{ \cos(a) }{1 + \sin(a) } + \frac{1 + \sin(a) }{ \cos(a) } = 2 \sec(a)$
give the answer
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Answer 1

Answer:

To prove:

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

Solution:

We can solve separately the LHS using trigonometric and algebric identities.

★Solving for LHS:

$\dfrac{\cos A}{1+\sin A} + \dfrac{1+\sin A}{\cos A}$

✦Taking LCM:

$\dashrightarrow$$\dfrac{(\cos A)^2 + (1+\sin A)^2}{(1+\sin A)(\cos A)}$

$\dashrightarrow$$\dfrac{\cos^2A + \sin^2A + 1 + 2\sin A}{(\cos A)(1+\sin A)}$

$\dashrightarrow$$\dfrac{1 +1+2\sin A}{(\cos A)(1+\sin A)}$

$\dashrightarrow$$\dfrac{2+2\sin A}{(1+\sin A)(\cos A)}$

$\dashrightarrow$$\dfrac{2(1+\sin A)}{(1+\sin A)(\cos A)}$

$\dashrightarrow$$\dfrac{2}{\cos A}$

$\dashrightarrow$$2 \sec A$

★Identities used:

$\rightarrow$1️⃣$sin^2A + cos^2A = 1$

$\rightarrow$2️⃣$\dfrac{1}{cosA} = secA$