[tex]\color{red} {{{\Large {\bf{To\:\:Simplify\::\frac{\sin ^4(x)-\cos ^4(x)}{\sin ^2(x)-\cos ^2(x)}}}}}}[/tex]
[tex]\color{green}{{{\large {\bf{Your\:\:Answer\::\frac{\sin ^4(x)-\cos ^4(x)}{\sin ^2(x)-\cos ^2(x)}=1}}}}}[/tex]
[tex]\color{yellow} {\Huge {\sf{Solution:}}}[/tex]
[tex]\color{blue} {\large {\bf{Factor\:\sin ^4(x)-\cos ^4(x)}}}[/tex]
[tex]\tt \color{blue} {\mathrm{Rewrite\:}\sin ^4(x)-\cos ^4(x)\mathrm{\:as\:}(\sin ^2(x))^2-(\cos ^2(x))^2=(\sin ^2(x))^2-(\cos ^2(x))^2}[/tex]
[tex]\color{fuchsia} {\normalsize {\mathrm{Apply\:exponent\:rule}:\quad \:a^{bc}=(a^b)^c}}[/tex]
[tex]\color{fuchsia} {\normalsize \sin ^4(x)=(\sin ^2(x))^2}[/tex]
[tex]\color{fuchsia} {\normalsize =(\sin ^2(x))^2-\cos ^4(x)}[/tex]=
[tex]\color{fuchsia} {\normalsize \mathrm{Apply\:exponent\:rule}:\quad \:a^{bc}=(a^b)^c}[/tex]
[tex]\color{fuchsia} {\normalsize \cos ^4(x)=(\cos ^2(x))^2}[/tex]
[tex]\color{fuchsia} {\normalsize =(\sin ^2(x))^2-(\cos ^2(x))^2}[/tex]=
[tex]\mathrm{Apply\:Difference\:of\:Two\:Squares\:Formula:\:}[/tex]x^2-y^2=(x+y)(x-y)
[tex](\sin ^2(x))^2-(\cos ^2(x))^2=(\sin ^2(x)+\cos ^2(x))(\sin ^2(x)-\cos ^2(x))[/tex]
=[tex](\sin ^2(x)+\cos ^2(x))(\sin ^2(x)-\cos ^2(x))[/tex]=
[tex]\color{blue} {\large {\bf{Factor\:\sin ^2(x)-\cos ^2(x)}}}[/tex]
[tex]\mathrm{Apply\:Difference\:of\:Two\:Squares\:Formula:\:}[/tex]x^2-y^2=(x+y)(x-y)
[tex]\sin ^2(x)-\cos ^2(x)=(\sin (x)+\cos (x))(\sin (x)-\cos (x))[/tex]
(x)=(sin(x)+cos(x))(sin(x)−cos(x))
=(\sin (x)+\cos (x))(\sin (x)-\cos (x))=(sin(x)+cos(x))(sin(x)−cos(x))
[tex]\large=(\sin ^2(x)+\cos ^2(x))(\sin (x)+\cos (x))(\sin (x)-\cos (x))
(x))(sin(x)+cos(x))(sin(x)−cos(x))
\large =\frac{(\sin ^2(x)+\cos ^2(x))(\sin (x)+\cos (x))(\sin (x)-\cos (x))}{\sin ^2(x)-\cos ^2(x)}[/tex]=
[tex]\mathrm{Apply\:Difference\:of\:Two\:Squares\:Formula:\:}[/tex]x^2-y^2=(x+y)(x-y)
[tex]\sin ^2(x)-\cos ^2(x)=(\sin (x)+\cos (x))(\sin (x)[/tex]
[tex] (x)=(sin(x)+cos(x))(sin(x)−cos(x))
=\frac{(\sin ^2(x)+\cos ^2(x))(\sin (x)+\cos (x))(\sin (x)-\cos (x))}{(\sin (x)+\cos (x))(\sin (x)-\cos (x))}[/tex]=
[tex]\mathrm{Cancel\:}\frac{(\sin ^2(x)+\cos ^2(x))(\sin (x)+\cos (x))(\sin (x)-\cos (x))}{(\sin (x)+\cos (x))(\sin (x)-\cos (x))}:\quad \sin ^2(x)+\cos ^2(x)Cancel
(sin(x)+cos(x))(sin(x)−cos(x))[/tex]
[tex]\mathrm{Cancel\:the\:common\:factor:}\:\sin (x)+\cos(x)Cancelthecommonfactor:sin(x)+cos(x)[/tex]
=[tex]\frac{(\sin ^2(x)+\cos ^2(x))(\sin (x)-\cos (x))}{\sin (x)-\cos (x)}[/tex]=
[tex]\mathrm{Cancel\:the\:common\:factor:}\:\sin (x)-\cos[/tex]
[tex]\mathrm{Use\:the\:following\:identity}:\quad \cos ^2(x)+\sin[/tex]
[tex]\huge \boxed{\color{red} {\ \huge =1}}[/tex]
