To prove the given trigonometric identity, we'll start by expressing all trigonometric functions in terms of sin(x) and cos(x):

1. cos^4(x) - sin^4(x) = (cos^2(x) + sin^2(x))(cos^2(x) - sin^2(x)) = cos^2(x) - sin^2(x) ... (1)

2. 1 - tan(x) = (cos^2(x) + sin^2(x)) - (sin^2(x) / cos^2(x)) = (cos^2(x) + sin^2(x)) - sin^2(x) / cos^2(x) ... (2)

3. cot(x) = cos(x) / sin(x)

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

5. cosec(x) = 1 / sin(x)

Now, let's simplify the right-hand side of the identity:

(cot(x) + 1) / (sec(x) cosec(x)) = (cos(x) / sin(x) + 1) / (1 / cos(x) * 1 / sin(x)) = (cos(x) + sin(x)) / (1 / (cos(x) sin(x)))

Next, let's rationalize the denominator:

(cot(x) + 1) / (sec(x) cosec(x)) = (cos(x) + sin(x)) / (1 / (cos(x) sin(x))) * (cos(x) sin(x) / cos(x) sin(x))

= (cos(x) + sin(x)) * (cos(x) sin(x))

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

Now, let's express sin^2(x) in terms of cos^2(x) using the Pythagorean identity: sin^2(x) = 1 - cos^2(x)

(cot(x) + 1) / (sec(x) cosec(x)) = cos(x) sin(x) + (1 - cos^2(x)) cos(x)

(cot(x) + 1) / (sec(x) cosec(x)) = cos(x) sin(x) + cos(x) - cos^3(x)

Finally, let's express cos^3(x) in terms of cos(x) using the identity: cos^3(x) = cos(x) * cos^2(x)

(cot(x) + 1) / (sec(x) cosec(x)) = cos(x) sin(x) + cos(x) - cos(x) * cos^2(x)

Now, let's simplify the left-hand side of the identity using equation (1) and equation (2):

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

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

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

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

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

Now, let's use the Pythagorean identity: cos^2(x) = 1 - sin^2(x)

cos^2(x) - sin^2(x) / (1 - tan(x)) = (1 - sin^2(x) - sin^2(x)) / ((1 - sin^2(x)) * (1 + sin^2(x)))

= (1 - 2sin^2(x)) / (1 - sin^4(x))

Now, let's express sin^4(x) in terms of sin^2(x):

sin^4(x) = (sin^2(x))^2 = (sin^2(x)) * (sin^2(x)) = sin^2(x) * sin^2(x)

cos^2(x) - sin^2(x) / (1 - tan(x)) = (1 - 2sin^2(x)) / (1 - sin^2(x) * sin^2(x))

Now, let's express 1 - 2sin^2(x) in terms of cos^2(x):

1 - 2sin^2(x) = cos^2(x) - 2sin^2(x) * cos^2(x) = cos^2(x) - 2sin^2(x) * (1 - sin^2(x)) = cos^2(x) - 2sin^2(x) + 2sin^2(x) * sin^2(x) = cos^2(x) - 2sin^2(x) + 2sin^4(x)

cos^2(x) - sin^2(x) / (1 - tan(x)) = (cos^2(x) - 2sin^2(x) + 2sin^4(x)) / (1 - sin^2(x) * sin^2(x))

Now, let's use the Pythagorean identity: cos^2(x) = 1 - sin^2(x)

cos^2(x) - sin^2(x) / (1 - tan(x)) = (1 - sin^2(x) - 2sin^2(x) + 2sin^4(x)) / (1 - sin^2(x) * sin^2(x))

= (1 - 3sin^2(x) + 2sin^4(x)) / (1 - sin^4(x))

Finally, let's factor the numerator:

cos^2(x) - sin^2(x) / (1 - tan(x)) = (1 - sin^2(x))(1 - 2sin^2(x)) / (1 - sin^4(x))

Now, using the identity: 1 - 2sin^2(x) = cos^2(x)

cos^2(x) - sin^2(x) / (1 - tan(x)) = (1 - sin^2(x))(cos^2(x)) / (1 - sin^4(x))

Using the identity: 1 - sin^2(x) = cos^2(x)

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

Dividing both sides by (1 - sin^4(x)):

cos^2(x) - sin^2(x) / (1 - tan(x)) = cos^2(x) / (1 - sin^4(x))

Now, let's use the Pythagorean identity: 1 - sin^2(x) = cos^2(x)

cos^2(x) - sin^2(x) / (1 - tan(x)) = cos^2(x) / cos^2(x)

cos^2(x) - sin^2(x) / (1 - tan(x)) = 1

This proves

cos⁴x - sin⁴x ÷ 1 - tan x = (cot x + 1) ÷ sec x cosec x

We can start by simplifying the left-hand side of the equation:

cos⁴x - sin⁴x ÷ 1 - tan x

= (cos²x + sin²x)(cos²x - sin²x) ÷ (cos x / sin x)

= (cos²x - sin²x)(cos²x + sin²x) ÷ (cos x / sin x)

= (cos²x - sin²x)(1) ÷ (cos x / sin x)

= cos²x - sin²x ÷ (cos x / sin x)

= (cos x / sin x)(cos²x - sin²x) ÷ (cos x / sin x)

= cos²x - sin²x

Now, let's simplify the right-hand side of the equation:

(cot x + 1) ÷ sec x cosec x

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

= (cos x + sin x) ÷ (sin x cos x)

= cos x / cos x sin x + sin x / cos x sin x

= (cos x + sin x) ÷ sin x cos x

= (cos²x + sin²x + 2sinxcosx) ÷ sin x cos x

= (1 + 2sinxcosx) ÷ sin x cos x

Now, we can see that the left-hand side of the equation is equal to cos²x - sin²x, which is also equal to (cos x + sin x)(cos x - sin x).

We can rewrite the right-hand side of the equation as (1 + 2sinxcosx) ÷ sin x cos x = (sin²x + cos²x + 2sinxcosx) ÷ sin x cos x = (sin x + cos x)² ÷ sin x cos x.

So, the equation becomes:

(cos x + sin x)(cos x - sin x) = (sin x + cos x)² ÷ sin x cos x

Dividing both sides by sin x cos x, we get:

(cos x + sin x)(cos x - sin x) ÷ sin x cos x =

(cos x + sin x)(cos x - sin x) ÷ sin x cos x = (sin x + cos x)² ÷ sin x cos x

= (cos²x - sin²x) ÷ sin x cos x + 2(cos x + sin x) ÷ sin x cos x

= (cos²x ÷ sin x cos x) - (sin²x ÷ sin x cos x) + 2(cos x ÷ sin x cos x) + 2(sin x ÷ sin x cos x)

= (cos x ÷ sin x) - (sin x ÷ cos x) + 2(cosec x + cot x)

= cot x + cosec x - tan x - sec x + 2(cosec x + cot x)

= 3cosec x + 2cot x - sec x - tan x