Can someone please explain how to simplify this proiblem:
cotx/sec^2 + cotx/csc^2
(cos/sin)/(1/ cos^2)+(cos/sin)/(1/sin^2)
cos^3/sin + cos sin
cos^3/sin + cos sin^2/sin
cos (cos^2 + sin^2)/sin
but cos^2 + sin^2 = 1
so
cos/sin
or
cot x
what happens to the cosine on the right side, when its in this step
cos^3x/sinx + cosxsinx^2/sinx
To simplify the problem, we can start by using trigonometric identities to convert the expressions given in terms of sine and cosine.
First, let's rewrite cot(x) in terms of cosine and sine:
cot(x) = cos(x)/sin(x)
Now, let's rewrite sec^2(x) in terms of cosine:
sec^2(x) = 1/cos^2(x)
Similarly, we can rewrite csc^2(x) in terms of sine:
csc^2(x) = 1/sin^2(x)
Plugging these identities back into the original problem, we get:
cos(x)/sin(x) * 1/cos^2(x) + cos(x)/sin(x) * 1/sin^2(x)
To simplify, let's find a common denominator for the two fractions. The common denominator will be sin^2(x) * cos^2(x):
(cos(x) * cos^2(x))/(sin(x) * sin^2(x)) + (cos(x) * sin^2(x))/(sin(x) * cos^2(x))
Expanding the numerators, the expression becomes:
(cos^3(x))/(sin^3(x)) + (cos(x) * sin^2(x))/(sin(x) * cos^2(x))
To add the fractions, we need a common denominator, which is sin^3(x) * cos^2(x):
[(cos^3(x) * cos^2(x)) + (cos(x) * sin^2(x) * sin^3(x))]/(sin^3(x) * cos^2(x))
Simplifying the numerators by combining like terms, we have:
(cos^5(x) + cos(x) * sin^5(x))/(sin^3(x) * cos^2(x))
And that's the simplified form of the given expression cot(x)/sec^2(x) + cot(x)/csc^2(x).