I would like to prove that
sinX^4 - cosX^4 + 2cosX^2
equals to 1 using sin^2 theta+ cos^2 theta=1 and a^2-b^2=(a-b)(a+b).
I am completely stuck though, and do not know where to begin. Any advice would be appreciated.
$\displaystyle \sin^4(x)-\cos^4(x)+2\cos^2(x)=1$
Let's focus on this term first:
$\displaystyle \sin^4(x)-\cos^4(x)$
using the formula $\displaystyle a^2-b^2=(a+b)(a-b)$, we see that
$\displaystyle \sin^4(x)-\cos^4(x)=(\sin^2(x)+\cos^2(x))(\sin^2(x)-\cos^2(x))=\sin^2(x)-\cos^2(x)$
Now, we see that the equation now becomes
$\displaystyle \sin^2(x)-\cos^2(x)+2\cos^2(x)=1$
Simplifying, we get
$\displaystyle \sin^2(x)+\cos^2(x)=1\implies 1=1$
Does this make sense?
--Chris