Question

Hi this is an equation from my pre-calc class and i don't really understand the steps in the conversion and factoring parts.

Answer 1

If you're just factorizing, you can do so by grouping.

cot³(x) + cot²(x) + cot(x) + 1

= cot²(x) (cot(x) + 1) + cot(x) + 1

= (cot²(x) + 1) (cot(x) + 1)

Put another way, if y = cot(x) + 1, then

cot²(x) y + y = (cot²(x) + 1) y

We can simplify this somewhat. Recall the Pythagorean identity,

sin²(x) + cos²(x) = 1

Dividing through both sides of the equation by sin²(x) reveals another form of the identity,

sin²(x)/sin²(x) + cos²(x)/sin²(x) = 1/sin²(x)

1 + cot²(x) = csc²(x)

Then we end up with

cot³(x) + cot²(x) + cot(x) + 1 = csc²(x) (cot(x) + 1)

Answer 2

Given:$cot^{3} x+cot^{2} x+cotx+1$

Factor:

$cot {}^{2} (x)(cot(x) + 1) + 1(cot(x) + 1$

$(cot {}^{2} (x) + 1)(cot(x) + 1)$

Substitute $\cot {}^{2} (x) + 1 = csc {}^{2} (x):csc {}^{2} (x)(cot(x) + 1)$