Reduce (csc^2 x - sec^2 X) to an expression containing only tan x.

(is this correct?)

csc x = 1/sin x
sec x = 1/cos x
tan x = 1/cot x

sin^2 x + cos^2 x = 1
1 + cot^2 x = csc^2 x
tan^2 x + 1 = sec^2 x

csc^2 x - sec^2 x
= 1 + cot^2 x - (1 + tan^2 x)
= cot^2 x - tan^2 x
= (1/tan^2 x) - tan^2 x

correct

thanks!

To reduce (csc^2 x - sec^2 x) to an expression containing only tan x, let's start by substituting the given values:

csc x = 1/sin x
sec x = 1/cos x
tan x = 1/cot x

Now, we know the following trigonometric identities:

sin^2 x + cos^2 x = 1
1 + cot^2 x = csc^2 x
tan^2 x + 1 = sec^2 x

Using these identities, let's simplify the expression:

csc^2 x - sec^2 x
= 1 + cot^2 x - (1 + tan^2 x) [Substituting the values for csc^2 x and sec^2 x]
= cot^2 x - tan^2 x [Simplifying the expression]

Now, to express the above expression only in terms of tan x, we know that:
cot x = 1/tan x [Defining the cotangent function]

Let's substitute this into the expression:

cot^2 x - tan^2 x
= (1/tan x)^2 - tan^2 x
= 1/tan^2 x - tan^2 x
= 1/tan^2 x - (tan^2 x * tan^2 x)
= 1/tan^2 x - tan^4 x

Therefore, reducing (csc^2 x - sec^2 x) to an expression containing only tan x gives us 1/tan^2 x - tan^4 x.