How do you simplify these equations:
1.)(2/cot^3(x) - cot^2 (x)) + (2/cot (x)-2)
2.) (2/ tan^2 (x)) + (2/ tan(x) -2)
To simplify the given equations, we will use trigonometric identities and some basic algebraic manipulations.
1. Simplifying (2/cot^3(x) - cot^2(x)) + (2/cot(x) - 2):
First, let's simplify the terms involving cot(x). Remember that cot(x) is equal to 1/tan(x):
Replacing cot(x) with 1/tan(x), we have:
(2/(1/tan^3(x)) - (1/tan^2(x))) + (2/(1/tan(x)) - 2)
Now, we can simplify the fractions:
(2 * tan^3(x) - tan^2(x)) + (2 * tan(x) - 2)
Next, we can distribute the coefficients through the parentheses:
2tan^3(x) - tan^2(x) + 2tan(x) - 2
2. Simplifying (2/tan^2(x)) + (2/tan(x) - 2):
Similar to the previous equation, let's simplify the terms involving tan(x):
(2/(tan^2(x))) + (2/tan(x) - 2)
Now, we can simplify the fractions:
(2/tan^2(x)) + (2/tan(x) - 2)
Next, we can combine the fractions under a common denominator:
(2 * tan(x) + 2 - 2tan^2(x)) / tan^2(x)
Lastly, we can rearrange the terms:
(2 - 2tan^2(x) + 2tan(x)) / tan^2(x)
These are the simplified forms of the given equations.