# Can (sec x - cosec x) / (tan x - cot x) be simplified further?

hala718 | Certified Educator

Given the expression ( sec x - csec x ) / (tan x - cot x)

We need to simplify.

We will use trigonometric identities to simplify.

We know that:

sec x = 1/cos x

csec x = 1/sin x

tan x = sinx/cosx

cot x = cos x/ sin x

We will substitute into the expression.

==> (1/cos x - 1/sin x ) / (sin x / cos x - cos x/ sin x)

==>[ ( sin x - cos x) / sinx*cosx] / [ (sin^2 x - cos^2 x)/ cosx*sinx

We will reduce similar:

==> (sinx - cos x) / (sin^2 x - cos^2 x)

Now we will simplify the denominator.

==> (sin x - cos x) / (sin x - cos x)(sinx + cos x)

Reduce similar terms.

==> 1/ (sin x + cos x)

Then the expression ( sec x - csec x)/(tanx - cot x) can be written as : 1/(sin x + cos x)

justaguide | Certified Educator

We have to simplify (sec x - cosec x) / (tan x - cot x) further.

(sec x - cosec x) / (tan x - cot x)

use sec x = 1/ cos x, cosec x = 1/ sin x , tan x = sin x / cos x and cot x = cos x / sin x

=> (1/cos x - 1/ sin x) / ((sin x/cos x) - (cos x / sin x))

=> [(sin x - cos x)/(cos x * sin x)] / [((sin x)^2 - (cos x)^2)/ (cos x * sin x)]

=> [(sin x - cos x)] / [(sin x)^2 - (cos x)^2]

=> (sin x - cos x) / (sin x - cos x)(sin x + cos x)

=> 1 / (sin x + cos x)

The required simplified form is 1 / (sin x + cos x)