4cot^2x-4/tanx+cosxsecx
write expression in factored form as an algebraic expression of a singletrig function
Use the following identities:
cot(x) = 1/tan(x)
sec(x) = 1/cos(x)
tan(x) = sin(x)/cos(x)
I suspect that parentheses are missing from the numerator and/or denominator.
The following problem may not be what you are working on. You can work out the problem using the above identities similar to the response below.
4cot²(x) - 4tan(x) + cos(x) sec(x)
=4/tan²(x) -4tan(x) + cos(x)/cos(x)
=4/tan²(x) -4tan(x) + 1
To write the expression 4cot^2x - 4/tanx + cosxsecx in factored form as an algebraic expression of a single trig function, we can simplify it using trigonometric identities.
First, let's simplify the numerator:
cot^2x = cos^2x / sin^2x
Thus, 4cot^2x = 4(cos^2x / sin^2x)
Now, let's simplify the denominator:
4/tanx = 4/(sinx/cosx) = 4*(cosx/sinx) = 4cosx/sinx
Now we can rewrite the expression:
4cot^2x - 4/tanx + cosxsecx
= 4(cos^2x / sin^2x) - 4cosx/sinx + cosxsecx
Next, let's rewrite secx in terms of cosx:
secx = 1/cosx
Now the expression becomes:
4(cos^2x / sin^2x) - 4cosx/sinx + cosx(1/cosx)
= 4cos^2x / sin^2x - 4cosx/sinx + 1
Now, let's find a common denominator for the first two terms in the numerator:
4cos^2x / sin^2x - (4cosx * sinx) / (sin^2x)
= (4cos^2x - 4cosxsinx) / sin^2x + 1
Factoring out a common factor of 4cosx from the numerator:
= 4cosx(cosx - sinx) / sin^2x + 1
Now, rewriting in terms of a single trig function:
= 4cosx(cosx - sinx) / sin^2x + 1
= 4cotxtanx + 1
Therefore, the expression 4cot^2x - 4/tanx + cosxsecx can be expressed as 4cotxtanx + 1, in factored form as an algebraic expression of a single trig function.
To write the expression in factored form as an algebraic expression of a single trig function, we can simplify the given expression first.
Let's rewrite the expression with additional parentheses to ensure clarity:
(4cot^2x - 4) / (tanx + cosxsecx)
We'll start by simplifying the numerator:
4cot^2x - 4
Using the Pythagorean identity for cotangent (cot^2x = 1 + tan^2x), we can substitute it in the expression:
4(1 + tan^2x) - 4
Simplifying further, we get:
4tan^2x
Now, let's simplify the denominator:
tanx + cosxsecx
Using the reciprocal identities (secx = 1/cosx), we can rewrite it as:
tanx + cosx / cosx
Combining like terms, we get:
(tanx + cosx) / cosx
Now, we can write the original expression as:
(4tan^2x) / [(tanx + cosx) / cosx]
To write it in factored form as an algebraic expression of a single trig function, we can simplify this further.
Recall the identity tanx = sinx / cosx:
(4(tanx)^2) / [(tanx + cosx) / cosx]
(4(sin^2x / cos^2x)) / [(tanx + cosx) / cosx]
The cos^2x terms cancel out:
4 sin^2x / (tanx + cosx)
Therefore, the expression in factored form as an algebraic expression of a single trig function is:
4 sin^2x / (tanx + cosx)