document.write( "Question 1107683: How to prove that $\"sinxcos%5E3x-cosxsin%5E3x=%281%2F4%29%28sin4x%29\"$ ? \n" ); document.write( "

Algebra.Com's Answer #722698 by rothauserc(4718) $\"\"$ $\"About$

You can put this solution on YOUR website!
sin(x)cos^3(x) - cos(x)sin^3(x) = (1/4) * sin(4x)
\n" ); document.write( ":
\n" ); document.write( "multiply both sides of = by 4
\n" ); document.write( ":
\n" ); document.write( "4(sin(x)cos^3(x) - cos(x)sin^3(x)) = sin(4x)
\n" ); document.write( ":
\n" ); document.write( "apply the identity cos^2(x) = 1-sin^2(x) to cos^3(x)
\n" ); document.write( ":
\n" ); document.write( "4(cos(x)(1-sin^2(x))sin(x) - cos(x)sin^3(x)) = sin(4x)
\n" ); document.write( ":
\n" ); document.write( "4((cos(x)-cos(x)sin^2(x))sin(x) - cos(x)sin^3(x)) = sin(4x)
\n" ); document.write( ":
\n" ); document.write( "4(cos(x)sin(x)-cos(x)sin^3(x) - cos(x)sin^3(x)) = sin(4x)
\n" ); document.write( ":
\n" ); document.write( "4(cos(x)sin(x)-2cos(x)sin^3(x)) = sin(4x)
\n" ); document.write( ":
\n" ); document.write( "4cos(x)sin(x)-8cos(x)sin^3(x) = sin(4x)
\n" ); document.write( ":
\n" ); document.write( "use double angle identity on sin(4x)
\n" ); document.write( ":
\n" ); document.write( "4cos(x)sin(x)-8cos(x)sin^3(x) = 2cos(2x)sin(2x)
\n" ); document.write( ":
\n" ); document.write( "use double angle identity on cos(2x)
\n" ); document.write( ":
\n" ); document.write( "4cos(x)sin(x)-8cos(x)sin^3(x) = 2(1-2sin^2(x))sin(2x)
\n" ); document.write( ":
\n" ); document.write( "use double angle identity on sin(2x)
\n" ); document.write( ":
\n" ); document.write( "4cos(x)sin(x)-8cos(x)sin^3(x) = 2 * 2cos(x)sin(x) * (1-2sin^2(x))
\n" ); document.write( ":
\n" ); document.write( "4cos(x)sin(x)-8cos(x)sin^3(x) = 4cos(x)sin(x) - 8cos(x)sin^3(x)
\n" ); document.write( ":
\n" ); document.write( "both sides are the same, the identity is correct
\n" ); document.write( ":
\n" ); document.write( " \n" ); document.write( "

\n" );