Question

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

Answer 1

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)`