rewrite sinxtanx/(cosx+1) as an expression that doesn't involve a fraction.
To rewrite sin(x)tan(x)/(cos(x)+1) without a fraction, we can use the concept of multiplying by the conjugate.
The conjugate of the denominator (cos(x)+1) is (cos(x)-1). Thus, we can multiply both the numerator and denominator by (cos(x)-1) to eliminate the fraction.
(sin(x)tan(x)(cos(x)-1))/(cos(x)+1)(cos(x)-1)
Expanding the numerator:
sin(x)tan(x)(cos(x)-1) = sin(x)tan(x)cos(x) - sin(x)tan(x)
Now, we have:
(sin(x)tan(x)cos(x)-sin(x)tan(x))/(cos^2(x)-1)
Using the trigonometric identity tan(x) = sin(x)/cos(x):
(sin(x)(sin(x)/cos(x))cos(x)-sin(x)(sin(x)/cos(x)))/(cos^2(x)-1)
Simplifying:
(sin^2(x)cos(x)/cos(x) - sin^2(x)/cos(x))/(cos^2(x)-1)
Now, we can cancel out the common factor in the numerator:
(sin^2(x) - sin^2(x)/cos(x))/(cos^2(x)-1)
Finally, we can simplify further:
(sin^2(x)(1 - 1/cos(x)))/(cos^2(x)-1)
Therefore, sin(x)tan(x)/(cos(x)+1) can be rewritten as sin^2(x)(1 - 1/cos(x))/(cos^2(x)-1).
To rewrite sin(x)tan(x)/(cos(x)+1) as an expression that doesn't involve a fraction, we need to simplify the expression by eliminating the fraction.
An approach to do this is to multiply both the numerator and the denominator of the fraction by the conjugate of the denominator, which in this case is (cos(x) - 1). This technique is called rationalizing the denominator.
First, let's rewrite the expression, multiplying the numerator and denominator by (cos(x) - 1):
(sin(x)tan(x)/(cos(x)+1)) * ((cos(x) - 1)/(cos(x) - 1))
Next, we can simplify the expression by distributing the numerator, and using the identity tan(x) = sin(x)/cos(x):
(sin(x) * sin(x) * cos(x) - sin(x) * tan(x))/(cos(x) * cos(x) - 1)
Now, let's simplify the numerator further:
(sin^2(x) * cos(x) - sin^2(x)/cos(x))/(cos^2(x) - 1)
Using the identity sin^2(x) = 1 - cos^2(x) and rearranging the terms, we get:
((1 - cos^2(x)) * cos(x) - (1 - cos^2(x)))/((cos(x))^2 - 1)
Simplifying further:
(cos(x) - cos^3(x) - 1 + cos^2(x))/((cos^2(x)) - 1)
Combining like terms, we get:
(cos(x) + cos^2(x) - cos^3(x) - 1)/(cos^2(x) - 1)
Therefore, sin(x)tan(x)/(cos(x)+1) can be rewritten as (cos(x) + cos^2(x) - cos^3(x) - 1)/(cos^2(x) - 1).