cot^5 x = cot x/sec^4 x - 2 sec^2 x +1

I tested for x = 10°

LS = cot^5 10° = 5866.87
RS = 4.27
LS ≠ RS

The way you typed it , the equation is NOT an identity

A good example of why we need to use parentheses.

Noticing that

sec^4 x - 2 sec^2 x +1 is a perfect square, let's try

RS:
cot x/(sec^4 x - 2 sec^2 x +1)
cot x/(sec^2 x - 1)^2
cot x/(tan^2 x)^2
cot x/tan^4 x
cot^5 x
= LS

To solve the equation cot^5 x = cot x/sec^4 x - 2 sec^2 x + 1, we will need to simplify both sides of the equation and find a common denominator.

Let's start by simplifying the right side of the equation. Since sec^2 x is the reciprocal of cos^2 x, we can rewrite it as 1/cos^2 x.

cot x/sec^4 x = cot x/(1/cos^4 x) = cos^4 x * cot x

Now, let's find a common denominator for the terms on the right side of the equation. The common denominator will be cos^4 x, so we need to rewrite 1 as cos^4 x / cos^4 x.

cot^5 x = cos^4 x * cot x - 2 sec^2 x + cos^4 x / cos^4 x

Next, we can combine the terms on the right side using a common denominator.

cot^5 x = cos^4 x * cot x - 2 sec^2 x + (cos^4 x / cos^4 x)

Now, simplify the equation further:

cot^5 x = cos^4 x * cot x - 2 sec^2 x + 1

Now that we have simplified the equation, we need to solve for cot x. To do that, let's rewrite cot^5 x as (cot x)^5:

(cot x)^5 = cos^4 x * cot x - 2 sec^2 x + 1

This equation is now in terms of cot x and sec x. To solve for cot x, we need to eliminate sec x. We can do that by using the identity: sec^2 x = 1 + tan^2 x.

Let's substitute sec^2 x with 1 + tan^2 x:

(cot x)^5 = cos^4 x * cot x - 2 (1 + tan^2 x) + 1

Now, let's simplify further:

(cot x)^5 = cos^4 x * cot x - 2 - 2tan^2 x + 1

(cot x)^5 = cos^4 x * cot x - 2tan^2 x - 1

At this point, we have an equation in terms of cot x and tan x. To solve for cot x, we will need to use additional trigonometric identities or numerical methods.