How do you verify csc (-x) / sec ( - x) =- cot x?

1 Answer
Jul 13, 2016

For this problem, we can make use of some trigonometric identities such as these:

sin(-x) = -sin(x)
cos(-x) = cos(x)
tan(-x) = -tan(x)

csc(x) = (1)/(sin(x))

sec(x) = (1)/(cos(x))

cot(x) = (1)/(tan(x)) = (cos(x))/(sin(x))

We can now try to rewrite our original equation, starting from the left-hand side, which gives us

Left-Hand Side:

(csc(-x))/(sec(-x)) = ((1)/(-sin x))/ ((1)/(cos x)) = ((1)/(-sinx))/ ((1)/(cancel(cos x))) *(cos x/1)/(cancel(cos x)/1) =- (cosx)/(sinx

Right-Hand Side:

-cot(x) = (1)/(-tan x) = - cos x / sin x

Since the left-hand side equals the right-hand side, we have verified that

csc(-x)/(sec(-x)) = -cot(x)