By definition, sec(x)= \frac{1}{cos(x)}sec(x)=1cos(x), and tan(x) = \frac{sin(x)}{cos(x)}tan(x)=sin(x)cos(x)
We can rewrite the fraction in parenthesis as
\frac{sec(x)}{tan(x)} = \frac{1}{cos(x)}\cdot \frac{cos(x)}{sin(x)} = \frac{1}{sin(x)}sec(x)tan(x)=1cos(x)⋅cos(x)sin(x)=1sin(x)
So, the expression becomes
1 - \frac{1}{sin^2(x)}1−1sin2(x)
To differentiate this expression, remember that
d/(dx) (1 - \frac{1}{sin^2(x)}) = d/(dx) (1) - d/(dx) (\frac{1}{sin^2(x)}) =- d/(dx) \frac{1}{sin^2(x)}ddx(1−1sin2(x))=ddx(1)−ddx(1sin2(x))=−ddx1sin2(x)
since the derivative of a number is zero.
Finally, you can write -1/sin^2(x)−1sin2(x) as -sin^{-2}(x)−sin−2(x), and derive it with chain rule:
d/(dx) -sin^{-2}(x) = -d/(dx) sin^{-2}(x) = - (-2sin^{-3}(x)*cos(x)) = \frac{2cos(x)}{sin^3(x)}ddx−sin−2(x)=−ddxsin−2(x)=−(−2sin−3(x)⋅cos(x))=2cos(x)sin3(x)
