(cosxcotx/secx+tanx)+(sinx/secx-tanx)
making everything sin and cos, we have
(cos^2/sin)/(1/cos + sin/cos) + sin/(1/cos - sin/cos)
cos^2/sin * cos/(1+sin) + sin*cos/(1-sin)
cos^3/(sin(1+sin)) + sin*cos/(1-sin)
(cos^3(1-sin) + sin^2*cos(1+sin))/(sin(1-sin^2))
(cos^3 - sin*cos^3 + cos*sin^2 + sin^3)/(sin*cos^2)
(cos^2 - sin*cos^2 + sin^2)/(sin*cos)
cot - cos + tan
whew - better double-check
To simplify the given expression, we can start by finding common denominators for the terms within the parentheses.
The first term, cos(x)cot(x)/sec(x) + tan(x), can be simplified as follows:
cos(x)cot(x)/sec(x) + tan(x)
= cos(x)(cos(x)/sin(x))/(1/cos(x)) + sin(x)/cos(x) (since cot(x) = cos(x)/sin(x) and sec(x) = 1/cos(x))
= cos^2(x)/sin(x) * cos(x) + sin(x)/cos(x)
= cos^3(x)/sin(x) + sin(x)/cos(x)
Similarly, the second term, sin(x)/sec(x) - tan(x), can be simplified as follows:
sin(x)/sec(x) - tan(x)
= sin(x)/(1/cos(x)) - sin(x)/cos(x) (since sec(x) = 1/cos(x))
= sin(x)*cos(x) - sin(x)/cos(x)
= sin(x)cos(x) - sin^2(x)/cos(x)
Combining both terms, we get:
(cos^3(x)/sin(x) + sin(x)/cos(x)) + (sin(x)cos(x) - sin^2(x)/cos(x))
Now, let's find a common denominator for both terms in the expression. The common denominator is sin(x)*cos(x):
(cos^3(x)*cos(x)/sin(x)*cos(x) + sin(x)*cos(x)/sin(x)*cos(x)) + (sin(x)cos(x)*sin(x)/sin(x)*cos(x) - sin^2(x)*sin(x)/sin(x)*cos(x))
Simplifying further:
= (cos^4(x) + sin(x)*cos^2(x))/sin(x)*cos(x) + (sin^2(x)*cos(x) - sin^3(x))/sin(x)*cos(x)
Now, we add the two fractions with the same denominator:
= (cos^4(x) + sin(x)*cos^2(x) + sin^2(x)*cos(x) - sin^3(x))/sin(x)*cos(x)
Rearranging the terms:
= (cos^4(x) + sin^2(x)*cos^2(x) + sin^2(x)*cos(x) - sin^3(x))/sin(x)*cos(x)
Finally, we can factor out a common factor:
= (cos^4(x) + sin^2(x)*cos^2(x) + sin^2(x)*cos(x) - sin^3(x))/(sin(x)*cos(x))
So, the simplified form of the given expression is (cos^4(x) + sin^2(x)*cos^2(x) + sin^2(x)*cos(x) - sin^3(x))/(sin(x)*cos(x)).