Rewrite cotx/(1+sinx) as an expression that does not involve a fraction.
To rewrite the expression cot(x) / (1 + sin(x)) without a fraction, we can use the property that dividing by a fraction is the same as multiplying by its reciprocal.
Recall that the reciprocal of a fraction is found by flipping the numerator and denominator. Therefore, the reciprocal of (1 + sin(x)) is (1 / (1 + sin(x))).
So, multiplying cot(x) by (1 / (1 + sin(x))) gives us:
cot(x) * (1 / (1 + sin(x)))
To simplify this further, let's rewrite cot(x) as cos(x) / sin(x):
(cos(x) / sin(x)) * (1 / (1 + sin(x)))
Now, let's combine the fractions by multiplying the numerators and denominators:
(cos(x) * 1) / (sin(x) * (1 + sin(x)))
Finally, we simplify further by distributing sin(x) to both terms in the denominator:
cos(x) / (sin(x) + sin^2(x))
Therefore, the expression cot(x) / (1 + sin(x)) can be rewritten as cos(x) / (sin(x) + sin^2(x)).
To rewrite the expression cot(x)/(1+sin(x)) without a fraction, we can multiply the numerator and denominator by the conjugate of (1+sin(x)).
The conjugate of (1+sin(x)) is (1-sin(x)).
So, multiplying the numerator and denominator by (1-sin(x)) gives us:
cot(x) * (1-sin(x)) / ((1+sin(x)) * (1-sin(x)))
Next, let's simplify the numerator:
cot(x) * (1- sin(x)) = cot(x) - cot(x) * sin(x)
Now, let's simplify the denominator:
(1+sin(x)) * (1-sin(x)) = 1 - sin^2(x)
Using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can rewrite sin^2(x) as 1 - cos^2(x):
1 - sin^2(x) = 1 - (1 - cos^2(x)) = cos^2(x)
Substituting this back into the denominator, we have:
1 - sin^2(x) = cos^2(x)
Therefore, the final expression without a fraction is:
(cot(x) - cot(x) * sin(x)) / cos^2(x)