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.