Question 1189605

Prove that 4(sin^4(x)+cos^4(x))=4-2sin^2(2x)

manipulate left side


{{{4(sin^4(x)+cos^4(x))}}}


={{{4(sin^2(x)sin^2(x) +cos^2(x)cos^2(x)  )}}}


={{{4(sin^2(x)(1-cos^2(x)) +(1-sin^2(x))cos^2(x) ) }}}


={{{4(sin^2(x)-sin^2(x)cos^2(x) +cos^2(x)-sin^2(x)cos^2(x) ) }}}


={{{4(sin^2(x)-2sin^2(x)cos^2(x) +cos^2(x) )}}} ..........{{{sin^2(x)+cos^2(x)=1}}}


={{{4(1-2sin^2(x)cos^2(x)  )}}} .............{{{sin^2(x)cos^2(x)  =1/4 sin^2(2 x)}}}


={{{4(1-2(1/4) sin^2(2 x)  ) }}}


={{{4-4*2(1/4) sin^2(2 x)  ) }}}


={{{4-2sin^2(2x)}}}  -> proven