How do I simplify integrating factor?

Question

I have obtained following integrating factor from ODE ($y^{\prime} + a(x) = g(x)$):

$ e^{\,A(x)} \:\:=\:\ e^{- \ln |\cos(x)|+c}$

where $ e^{\,A(x)}$ is integrating factor and $A(x) = \int a(x)\,\mathrm{d}x$.

$e^{- \ln |\cos(x)|+c}$ should equal to $ \:\frac{1}{\cos x}\:$.

How do I simplify the equation to $ \:\frac{1}{\cos x}\:$?

Thanks. ⁡

Answer 1

Use that $$-\log(\cos(x))=\log\frac{1}{\cos(x)}$$