how to simplify sqrt(8cos^4(t)-16cos^2(t)sin^2(t)+8sin^4(t))

[acpodgorski]

I figure i need to use some trig identities but I can't figure out which ones...
can i use cos^2+sin^2=1 somehow?

simplify:

sqrt(8cos^4(t) -16cos^2(t)sin^2(t)+8sin^4(t))

 

[galactus]

First factor out 8:

\(\displaystyle \L\\\sqrt{8(cos^{4}t-2cos^{2}tsin^{2}t+sin^{4}t}\)

Now factor the inside:

\(\displaystyle \L\\\sqrt{8(cost+sint)^{2}(cost-sint)^{2}}\)

Now, finish?.

 

[acpodgorski]

can i take the square root of each piece inside since they are being multiplied?

 

[acpodgorski]

and also is that the correct factoring? because i think that would mean that the sin^4t would have to be negative

 

[galactus]

Yes, that's the correct factoring.

It's the same as factoring \(\displaystyle \L\\x^{4}-2x^{2}y^{2}+y^{4}=(x-y)^{2}(x+y)^{2}\)

\(\displaystyle \L\\\sqrt{8(cost-sint)^{2}(cost+sint)^{2})}\)

\(\displaystyle \L\\\sqrt{8}(cost-sint)(cost+sint)\)

Notice the difference of two squares?.

\(\displaystyle \L\\\sqrt{8}(cos^{2}t-sin^{2}t)\)

Now, remeber your identites. Can you simplify further?.