Answer
$$ - 1$$
Work Step by Step
$$\eqalign{
& \int_{ - \pi /3}^0 {\sec x\tan x} dx \cr
& {\text{We know that}}\cr
&\,\,\,\frac{d}{{d\theta }}\left[ {\sec \theta } \right] = \sec \theta \tan \theta \cr
&{\text{Then}}{\text{,}} \cr
& \int_{ - \pi /3}^0 {\sec x\tan x} dx = \left( {\sec x} \right)_{ - \pi /3}^0 \cr
& {\text{Evaluate}} \cr
& \left( {\sec x} \right)_{ - \pi /3}^0 = \sec \left( 0 \right) - \sec \left( { - \frac{\pi }{3}} \right) \cr
& = \frac{1}{{\cos \left( 0 \right)}} - \frac{1}{{\cos \left( { - \pi /3} \right)}} \cr
& {\text{Simplifying}} \cr
& = \frac{1}{1} - \frac{1}{{1/2}} \cr
& = 1 - 2 \cr
& = - 1 \cr} $$
