Please solve it in step by step.

integrate: ∫(cot2x-csc2x)^2 dx

∫(cot2x-csc2x)^2 dx

∫ cot^2(2x) - 2cot(2x)csc(2x) + csc^2(2x) dx
∫ 2csc^2(2x) - 1 - 2cos(2x)/sin^2(2x) dx

Now everything is standard forms, so you get

-cot(2x) - x + 1/sin(2x)
csc(2x) - cot(2x) - x

or, you can go on to get

(1-cos(2x))/sin(2x) - x
(1 - 2cos^2(x) + 1)/(2sinx cosx) - x
(2 - 2cos^2(x))/(2sinx cosx) - x
sinx/cosx - x
tanx - x

+ C!

OR

cot ( 2x ) - csc ( 2x ) = cos ( 2x ) / sin ( 2x ) - 1 / sin ( 2x ) =

[ cos ( 2x ) - 1 ) ] / sin ( 2x )

Since the:

cos ( 2x ) = cos ^ 2 x - sin ^ 2 x

sin ( 2 x ) = 2 sin x cos x

you can write:

cot ( 2x ) - csc ( 2x ) = ( cos ^ 2 x - sin ^ 2 x - 1 ) / 2 sin x cos x =

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

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

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

- 2 sin ^ 2 x / 2 sin x cos x =

- 2 sin x * sin x / 2 sin x cos x =

- sin x / cos x = - tan x

So:

cot ( 2x ) - csc ( 2x ) = - tan x

Now:

[ cot ( 2x ) - csc ( 2x ) ] ^ 2 = ( - tan x ) ^ 2

[ cot ( 2x ) - csc ( 2x ) ] ^ 2 = tan ^ 2 x

�ç [ cot ( 2x ) - csc ( 2x ) ] ^ 2 dx = �ç tan ^ 2 x dx =

�ç ( sin ^ 2 x / cos ^ 2 x ) dx =

�ç [ ( 1 - cos ^ 2 x ) / cos ^ 2 x ] dx =

�ç [ ( 1 / cos ^ 2 x - cos ^ 2 x / cos ^ 2 x ) ] dx =

�ç ( sec ^ 2 x - 1 ) dx =

�ç sec ^ 2 x dx - �ç 1 * dx =

�ç sec ^ 2 x dx - �ç dx =

tan x - x + C

I like it!

But the half-angle formula would have gone directly to
(cos2x - 1)/sin2x = -tanx

Then we'd have

∫tan^2(x) dx
= ∫sec^2(x)-1 dx
= tanx - x + C

To integrate the expression ∫(cot^2(x) - csc^2(x))^2 dx, we'll take it step by step.

Step 1: Expand the square
∫(cot^2(x) - csc^2(x))^2 dx
= ∫(cot^4(x) - 2cot^2(x)csc^2(x) + csc^4(x)) dx

Step 2: Simplify the integral
∫(cot^4(x) - 2cot^2(x)csc^2(x) + csc^4(x)) dx
= ∫cot^4(x) dx - 2∫cot^2(x)csc^2(x) dx + ∫csc^4(x) dx

Step 3: Integrate each term separately
The three integrals can be solved using basic integral rules:

∫cot^4(x) dx:
We can rewrite cot^4(x) as (cot^2(x))^2 and use the substitution u = cot(x), du = -csc^2(x) dx:
= ∫u^2 (-1/2) du
= -1/2 ∫u^2 du
= -1/2 * (u^3/3) + C
= -1/6 cot^3(x) + C

∫2cot^2(x)csc^2(x) dx:
We can rewrite 2cot^2(x)csc^2(x) as 2(cot(x)csc(x))^2 and use the substitution u = cot(x)csc(x), du = -csc^2(x) dx:
= -2 ∫u^2 du
= -2 (u^3/3) + C
= -2/3 (cot^3(x)csc^3(x)) + C

∫csc^4(x) dx:
We can rewrite csc^4(x) as (csc^2(x))^2 and use the substitution u = csc(x), du = -csc(x)cot(x) dx:
= -∫u^2 (1/u) du
= -∫u du
= - (u^2/2) + C
= -1/2 csc^2(x) + C

Step 4: Combine the results
Substituting the results back into the original expression, we get:
∫(cot^2(x) - csc^2(x))^2 dx
= (-1/6 cot^3(x) + C) - 2/3 (cot^3(x)csc^3(x)) + (-1/2 csc^2(x) + C)
= -1/6 cot^3(x) - 2/3 cot^3(x)csc^3(x) - 1/2 csc^2(x) + C

So, the integral of (cot^2(x) - csc^2(x))^2 dx is equal to -1/6 cot^3(x) - 2/3 cot^3(x)csc^3(x) - 1/2 csc^2(x) + C, where C is the constant of integration.