Question

I need to write cos^2(x)+2cos(x)+1 in factored form including only one function. The answer is (cos(x)+1)^2 but how would you get to that answer?

Answer 1

\[
\begin{split}
&\cos^2(x)+2\cos(x)+1\\
=&\cos^2(x)+\cos(x)+\cos(x)+1\\
=&\cos(x)(\cos(x)+1)+1(\cos(x)+1)\\
=&(\cos(x)+1)^2
\end{split}
\]

Answer 2

So one way, is splitting the middle.

Answer 3

Let's think of cos in terms of x for a second.
Instead of:
\[\cos ^{2}x+2cosx+1\]
We could write a similar equation:
\[x ^{2}+2x+1\]
When we factor this we get:
(x+ )(x+ )
a=1 times c=1 gives us 1. Factors of one that equal two are one and one (woah).
so,
(x+1)(x+1)
is the factored form of:
\[x ^{2}+2x+1\]

Now, for \[\cos ^{2}x+2cosx+1\]
we follow the same principles.
(cosx+ )(cosx+ )
(cosx+1)(cosx+1)
or
(cosx+1)^2

Answer 4

Is this making sense now?

Answer 5

Yes thank you so much. I tried to get help from my parents, but I guess I've surpassed them in their mathematical knowledge.

Answer 6

One question of you're still there. Does this strategy work once other functions are thrown into the mix, or would I have to do something else?

Answer 7

It works best if you have all the same function.

Answer 8

Okay thanks.

Answer 9

Usually you can use trigonometric identities to perform substitutions. If that's the case, go for it. You might be able to cancel out a function and put it in terms of a single function.
Just be suspicious of 1-2sinx, 1, cos^2x + sin^2x, 1-sin^2x, 1-cos^2x. The same goes for the other pythagorean identities.

Answer 10

You know, tan^2x+1=sec^2x
and 1+cot^2x=csc^2x

Answer 11

Yeah:) Thank you so much, this was a life saver.

Answer 12

Glad to hear that :D If you have any more trig questions then feel free to ask!