document.write( "Question 625685: if sin2x=4sin2y ,show that (5tan (x-y)=3tan(x+y)) \n" ); document.write( "
Algebra.Com's Answer #393714 by Edwin McCravy(20065)\"\" \"About 
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document.write( "sin(2x) = 4·sin(2y), show that 5·tan(x-y) = 3·tan(x+y)\r\n" );
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document.write( "We will show that both equations are equivalent to the same equation.\r\n" );
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document.write( "                                  sin(2x) = 4·sin(2y)\r\n" );
document.write( "                          2·sin(x)·cos(x) = 4·[2sin(y)·cos(y)]\r\n" );
document.write( "                          2·sin(x)·cos(x) = 8·sin(y)·cos(y)\r\n" );
document.write( "                            sin(x)·cos(x) = 4·sin(y)·cos(y)\r\n" );
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document.write( "Now we will work with the second equation:\r\n" );
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document.write( "                               5·tan(x-y) = 3·tan(x+y)\r\n" );
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document.write( "                                  \"5%2Aexpr%28sin%28x-y%29%2Fcos%28x-y%29%29\" = \"3%2Aexpr%28sin%28x%2By%29%2Fcos%28x%2By%29%29\"\r\n" );
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document.write( "                                   \"expr%285sin%28x-y%29%2Fcos%28x-y%29%29\" = \"expr%283sin%28x%2By%29%2Fcos%28x%2By%29%29\"\r\n" );
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document.write( "                       5·sin(x-y)cos(x+y) = 3·sin(x+y)cos(x-y)\r\n" );
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document.write( "5·[sin(x)cos(y)-cos(x)sin(y)][cos(x)cos(y)-sin(x)sin(y)] =\r\n" );
document.write( "                  3·[sin(x)cos(y)+cos(x)sin(y)][cos(x)cos(y)+sin(x)sin(y)]\r\n" );
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document.write( "Since this is very bulky, I will use abbreviations:\r\n" );
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document.write( "capital S = sin(x), capital C = cos(x), small s = sin(y), small c = cos(y)\r\n" );
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document.write( "                          5[Sc-Cs][Cc-Ss] = 3[Sc+Cs][Cc+Ss]\r\n" );
document.write( "                   5[SCc²-S²sc-C²sc+CSs²] = 3[SCc²+S²sc+C²sc+CSs²]\r\n" );
document.write( "                   5[SCc²-(S²+C²)sc+CSs²] = 3[SCc²+(S²+C²)sc+CSs²]\r\n" );
document.write( "                   5[SCc²+CSs²-(S²+C²)sc] = 3[SCc²+CSs²+(S²+C²)sc]\r\n" );
document.write( "                   5[(c²+s²)SC-(S²+C²)sc] = 3[(c²+s²)SC+(S²+C²)sc]                     \r\n" );
document.write( "                           5[(1)SC-(1)sc] = 3[(1)SC+(1)sc]\r\n" );
document.write( "                                 5[SC-sc] = 3[SC+sc]\r\n" );
document.write( "                                 5SC-5sc  = 3SC+3sc\r\n" );
document.write( "                                      2SC = 8sc\r\n" );
document.write( "                                       SC = 4sc\r\n" );
document.write( "                            sin(x)·cos(x) = 4·sin(y)·cos(y)\r\n" );
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document.write( "So the equations are equivalent.\r\n" );
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document.write( "Edwin

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