True or False
The left side of cos^4 θ + (cos^2 θ)(sin^2 θ) + sin^2 θ = 1 can be factored to cos^2 θ(cos^2 θ + sin^2 θ) + sin^2 θ.
I would strongly suggest that you expand
cos^2 θ(cos^2 θ + sin^2 θ) + sin^2 θ
and see what you get.
I got 1. Thank you!
The factoring is correct, but it's not 1.
Every time I expand it I get 1.
you are missing a factor of 2.
(cos^2 θ + sin^2 θ)^2 = cos^4 θ + 2(cos^2 θ)(sin^2 θ) + sin^4 θ = 1
So it's false?
NO!
The left side factors as indicated.
The problem is, that it does not equal 1.
Maybe you typed the question wrong.
Oh, well I copied it directly from the assignment. Maybe there was a mistake made there.
False.
To determine if the left side of the equation can be factored to cos^2 θ(cos^2 θ + sin^2 θ) + sin^2 θ, we need to simplify both expressions and see if they are equal.
First, let's simplify the left side of the equation:
cos^4 θ + (cos^2 θ)(sin^2 θ) + sin^2 θ.
Notice that we have a common factor of cos^2 θ in the first two terms. We can factor it out:
cos^2 θ(cos^2 θ + sin^2 θ) + sin^2 θ.
Now, the expression inside the parenthesis is cos^2 θ + sin^2 θ. According to the Pythagorean identity, this expression is equal to 1:
cos^2 θ(1) + sin^2 θ.
Therefore, the simplified expression is:
cos^2 θ + sin^2 θ.
Using the Pythagorean identity again, we know that cos^2 θ + sin^2 θ is equal to 1.
Hence, the simplified left side of the equation is 1, not cos^2 θ(cos^2 θ + sin^2 θ) + sin^2 θ.
Therefore, the statement is false.