document.write( "Question 604347: verify the identity
\n" ); document.write( "(cos(x)+isin(x))^3=cos(3x)+isin(3x)\r
\n" ); document.write( "\n" ); document.write( "so far i have completed the following
\n" ); document.write( "(cos(x)+isin(x))+(cos(x)+isin(x))+(cos(x)+isin(x))
\n" ); document.write( "(cos^2(x)+isin(x)cos(x)+isin(x)cos(x)+(-1)sin^2(x))(cos(x)+isin(x))
\n" ); document.write( "(cos^2(x)+2isin(x)cos(x)-sin^2(x))(cos(x)+isin(x))
\n" ); document.write( "cos^3(x)+isin(x)cos^2(x)+2isin(x)cos^2(x)+2(-1)sin^2(x)cos(x)-sin^2(x)cos(x)-isin^3(x)
\n" ); document.write( "cos^3(x)+3isin(x)cos^2(x)-3sin^2(x)cos(x)-isin^3(x) \n" ); document.write( "
Algebra.Com's Answer #381121 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "\r\n" );
document.write( "It's immediate by DeMoivre's theorem, but I suppose they want\r\n" );
document.write( "you to prove it from scratch without using DeMoivre's theorem.  \r\n" );
document.write( "OK, here's how I approach it:\r\n" );
document.write( "\r\n" );
document.write( "[cos(x) + i·sin(x)]³ =\r\n" );
document.write( "\r\n" );
document.write( "[cos(x) + i·sin(x)]²[cos(x) + i·sin(x)]\r\n" );
document.write( "\r\n" );
document.write( "We'll work on the left bracketed expression only for awhile,\r\n" );
document.write( "and just keep bringing down the right bracket expression\r\n" );
document.write( "until we get the left bracket simplified:\r\n" );
document.write( "\r\n" );
document.write( "[cos²(x) + 2i·cos(x)sin(x)+ i²·sin²(x)][cos(x) + i·sin(x)]\r\n" );
document.write( "\r\n" );
document.write( "[cos²(x) + 2i·cos(x)sin(x)+ (-1)·sin²(x)][cos(x) + i·sin(x)]\r\n" );
document.write( "\r\n" );
document.write( "[cos²(x) + 2i·cos(x)sin(x) - sin²(x)][cos(x) + i·sin(x)]\r\n" );
document.write( "\r\n" );
document.write( "[{cos²(x) - sin²(x)} + i·{2cos(x)sin(x)}][cos(x) + i·sin(x)]\r\n" );
document.write( "\r\n" );
document.write( "[{cos²(x) - sin²(x)} + i·{2sin(x)cos(x)}][cos(x) + i·sin(x)]\r\n" );
document.write( "\r\n" );
document.write( "Use the identities:\r\n" );
document.write( "\r\n" );
document.write( "cos²(A) - sin²(A) = cos(2A)\r\n" );
document.write( "2sin(A)cos(A) = sin(2A)\r\n" );
document.write( "\r\n" );
document.write( "[cos(2x) + i·sin(2x)][cos(x) + i·sin(x)] =\r\n" );
document.write( "\r\n" );
document.write( "Now we finally use the right bracket and multiply out:\r\n" );
document.write( "\r\n" );
document.write( "cos(2x)cos(x) + i·cos(2x)sin(x) + i·sin(2x)cos(x) + i²·sin(2x)cos(x) =\r\n" );
document.write( "\r\n" );
document.write( "cos(2x)cos(x) + i·[cos(2x)sin(x) + sin(2x)cos(x)] + (-1)·sin(2x)cos(x) =\r\n" );
document.write( "\r\n" );
document.write( "cos(2x)cos(x) + i·[cos(2x)sin(x) + sin(2x)cos(x)] - sin(2x)cos(x) =\r\n" );
document.write( "\r\n" );
document.write( "cos(2x)cos(x) - sin(2x)sin(x) + i·[cos(2x)sin(x) + sin(2x)cos(x)] =\r\n" );
document.write( "\r\n" );
document.write( "cos(2x)cos(x) - sin(2x)sin(x) + i·[sin(2x)cos(x) + cos(2x)sin(x)] =\r\n" );
document.write( "\r\n" );
document.write( "Use the identities: \r\n" );
document.write( "\r\n" );
document.write( "cos(A)cos(B) - sin(A)sin(B) = cos(A+B)\r\n" );
document.write( "sin(A)cos(B) + cos(A)sin(B) = sin(A+B)\r\n" );
document.write( "\r\n" );
document.write( "cos(2x+x) + i·sin(2x+x)\r\n" );
document.write( "\r\n" );
document.write( "cos(3x) + i·sin(3x)\r\n" );
document.write( "\r\n" );
document.write( "You can probably leave out some of those steps where I just regrouped\r\n" );
document.write( "terms.\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );