Hannah_Ahn 4 years ago 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

1. Hannah_Ahn

2. PaxPolaris

$(\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$ $\because \left( a-b \right)^2=\left( a+b \right)^2-4ab$

3. PaxPolaris

$=\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)$

4. PaxPolaris

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

5. Hannah_Ahn

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!!!!!