cos(x) (tan(x) + cot(x))
Recall that tan(x) = sin(x)/cos(x) and cot(x) = 1/tan(x) = cos(x)/sin(x). Then
cos(x) (sin(x)/cos(x) + cos(x)/sin(x)) = sin(x) + cos²(x)/sin(x)
Write both terms with a common denominator and combine the fractions:
sin²(x)/sin(x) + cos²(x)/sin(x) = (sin²(x) + cos²(x))/sin(x)
Recall the Pythagorean identity,
sin²(x) + cos²(x) = 1
so our expression reduces to
1/sin(x)
Finally, recall that csc(x) = 1/sin(x) and we're done. So
cos(x) (tan(x) + cot(x)) = csc(x)
