Simplify e^(- natural log of cos(x))

$e^{- \ln (\cos (x))}$

Simplify $- \ln (\cos (x))$ by moving $- 1$ inside the logarithm.

$e^{\ln (\cos^{- 1} (x))}$

Exponentiation and log are inverse functions.

$\cos^{- 1} (x)$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{\cos (x)}$

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\sec (x)$