the expression cot times sec is equivalent to what

the expression cot times sec is equivalent to what

Weierstrass substitution:

$$\\\small{\text{ $ \boxed{ \rm{if~}t = \tan\tfrac{x}{2} \rm{~then~} }$}}\\ \small{\text{ $ \boxed{ \sin x = \frac{2t}{1 + t^2},~ \cos x = \frac{1 - t^2}{1 + t^2},~ \tan x = \frac{2t}{1 - t^2},~ \cot x = \frac{1 - t^2}{2t},~ \sec x = \frac{1 + t^2}{1 - t^2},~ \csc x = \frac{1 + t^2}{2t}. }$}}$$

$$\cot{(x)} \cdot \sec {(x)} = \left( \frac{1 - t^2}{2t} \right) \cdot \left( \frac{1 + t^2}{1 - t^2} \right) = \frac{1 + t^2}{2t} = \csc{( x )}$$

cotx = cos x / sin x    and     sec x  =  1/cos x

So

cot x * sec x =   (cos x / sin x) * (1 / cos x)  =  1 / sin x =  csc x

cot(x)  =  cos(x) / sin(x)

sec(x)  =  1 / cos(x)

cot(x) · sec(x)  =  [ cos(x) / sin(x) ] · [ 1 / cos(x) ]  =  1/ sin(x)  =  csc(x)