Question

which expression is equivalent to
\[(\sin^2\beta-\cos^2\beta)^2 - \sin^22\beta\]?

1.-2sin^2beta 2. 2sin^2 2beta 3. -cos4beta 4. cos4beta

and the answer is 4.
but again, i don't know why

Answer 1

please help me figure out why ..
thanks.

Answer 2

\[(\sin^2\beta-\cos^2\beta)^2 - \sin^22\beta\]\[=\left[ (\sin^2\beta+\cos^2\beta)^2-4\sin^2\beta \cos^2\beta \right] - \sin^22\beta\]
\[\because \left( a-b \right)^2=\left( a+b \right)^2-4ab\]

Answer 3

\[=\left[ (1)^2-\left( 2\sin \beta \cos \beta \right)^2 \right] - \sin^22\beta\]\[=\left[ 1-\sin^2\left( 2 \beta \right) \right] - \sin^2(2\beta)\]

Answer 4

\[=\left[ \cos^2\left( 2\beta \right) \right]-\sin^2(2\beta)\]\[=\cos(4\beta)\]

Answer 5

WOW
You are brilliant!!!
Hahaha, I keep going back to your steps and I am still on a way to figure out how you got that...
but that really helps me alot! thanks!!!!!