1. cos(x) = 1 / sec(x)
2. cos(x) = 1 / sqrt[1 + tan^2(x)]
3. cos(x) = 1 / sqrt[1 + 1/cot^2(x)]
4. cos(x) = 1 / sqrt[(cot^2(x)/cot^2(x) + 1/cot^2(x)]
5. cos(x) = 1 / sqrt[(cot^2(x) + 1) / cot^2(x)]
6. cos(x) = 1 / [sqrt(cot^2(x) + 1) / sqrt(cot^2(x)]
7. cos(x) = 1 / [sqrt(cot^2(x) + 1) / cot(x)]
8. cos(x) = cot(x) / sqrt(cot^2(x) + 1)

Begin by writing cos(x) in terms of sec(x).

In step 2 we rewrite sec(x) in terms of tan(x), by the Pythagorean identity 1 + tan^2(x) = sec^2(x).  Since we need sec(x), not sec^2(x), we take the square root of both sides of the identity, which gives us sec(x) = sqrt[1 + tan^2(x)].  We plug this into sec(x) from step 1.

In step 3 we rewrite tan(x) in terms of cot(x).  Since cot(x) is the reciprocal of tan(x), tan^2(x) = 1/cot^2(x).

In step 4 we write cot^2(x)/cot^2(x) in place of 1, since this will give us a common denominator within the square root.

In step 5, with the common denominator cot^2(x), we are able to combine the numerators into one fraction.

In step 6, we break up the square root within the denominator.  In the numerator of the denominator, we have sqrt(cot^2(x) + 1).  In the denominator of the denominator, we have sqrt(cot^2(x)).

In step 7, we simplify the denominator of the denominator.  The square root of cot^2(x) is cot(x).

In step 8 we simplify.  Since we were dealing with a numerator and denominator that were themselves in a denominator, we bring cot(x), the denominator of the denominator, up to the numerator with 1.  Since 1*cot(x) is still cot(x), we can ignore the 1.  Cosine is now written in terms of cotangent.

Hope this helps!