Perform the indicated operation and simplify the result .
Cot x(tan x- sec x)
Please help , I am struggling a lot !! Thank you!!!
cot = 1 / tan = cos / sin
sec = 1 / cos
distributing ... (cot * tan) - (cot * sec)
1 - (1 / sin) = 1 - csc
To simplify the given expression Cot x(tan x - sec x), we can use trigonometric identities.
Recall that the cotangent function is the reciprocal of the tangent function:
cot(x) = 1/tan(x)
Using this identity, we can rewrite the expression as follows:
cot(x) * (tan(x) - sec(x))
Now let's expand the expression using the definitions of tangent and secant:
cot(x) * (sin(x)/cos(x) - 1/cos(x))
Next, we find a common denominator for the terms inside the parentheses:
cot(x) * [(sin(x) - 1) / cos(x)]
Finally, we multiply the numerator and denominator by cot(x) to get a single fraction:
[(sin(x) - 1) / cos(x)] * cot(x)
Expanding the expression further, we have:
(sin(x) * cot(x) - cot(x)) / cos(x)
Now, we replace cot(x) with its reciprocal 1/tan(x):
[(sin(x) * (1/tan(x))) - (1/tan(x))] / cos(x)
Simplifying this expression, we have:
[sin(x)/tan(x) - 1/tan(x)] / cos(x)
Now, we can simplify the fractions:
[sin(x) - 1] / sin(x) / cos(x)
To divide by a fraction, we can multiply by its reciprocal. In this case, we multiply by the reciprocal of sin(x)/cos(x), which is cos(x)/sin(x):
[sin(x) - 1] / sin(x) * cos(x)/sin(x)
Now, we can simplify further:
[sin(x) * cos(x) - 1] / sin(x)^2
This is the simplified expression for the given expression cot(x)(tan(x) - sec(x)).
To simplify the given expression, we need to apply the trigonometric identities and simplify further. Let's break it down step by step:
Given expression: Cot x(tan x - sec x)
1. Rewrite cotangent (cot x) as 1/tan x:
Cot x = 1/tan x
2. Distribute cot x to the terms inside the parentheses:
1/tan x * (tan x - sec x)
3. Simplify by canceling out common terms:
The tangent (tan x) cancels out with the 1/tan x, leaving us with:
1 - sec x
4. Rewrite secant (sec x) as 1/cos x:
1 - 1/cos x
5. Combine the fractions by finding a common denominator:
Multiply the numerator and denominator of 1 by cos x:
(cos x - 1) / cos x
So, the simplified form of Cot x(tan x - sec x) is (cos x - 1) / cos x.