document.write( "Question 117920: Verify the identity:
\n" ); document.write( " tan(2 tan^-1 x)= 2 tan (tan^-1 x + tan^-1 x^3) \n" ); document.write( "
Algebra.Com's Answer #86090 by Edwin McCravy(20060)\"\" \"About 
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document.write( "Verify the identity:\r\n" );
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document.write( "tan(2tan-1x)= 2 tan(tan-1x + tan-1x3)\r\n" );
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document.write( "Let tan-1x = A and let tan-1x3 = B\r\n" );
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document.write( "Then tanA = x and tanB = x3\r\n" );
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document.write( "and the identity to verify becomes:\r\n" );
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document.write( "tan(2A)= 2tan(A + B)\r\n" );
document.write( "                                                2tanA\r\n" );
document.write( "On the left side use the identity: tan(2A) = ——————————— \r\n" );
document.write( "                                              1 - tan2A\r\n" );
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document.write( "                                                 tanA + tanB\r\n" );
document.write( "On the right side use the identity: tan(A+B) = ————————————————\r\n" );
document.write( "                                                1 - tanA·tanB\r\n" );
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document.write( "   2tanA           tanA + tanB   \r\n" );
document.write( "———————————  = 2·————————————————\r\n" );
document.write( " 1 - tan²A        1 - tanA·tanB  \r\n" );
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document.write( "Since tanA = x and tanB = x3, the above becomes\r\n" );
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document.write( "      2x           x + x3   \r\n" );
document.write( "   ————————  = 2·——————————\r\n" );
document.write( "    1 - x2        1 - x·x3  \r\n" );
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document.write( "Factor the numerator on the right getting x(1 + x)\r\n" );
document.write( "Multiply x·x3 getting x4\r\n" );
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document.write( "      2x           x(1 + x2)   \r\n" );
document.write( "   ————————  = 2·————————————\r\n" );
document.write( "    1 - x2          1 - x4\r\n" );
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document.write( "or putting the 2 factor on the right in the numerator\r\n" );
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document.write( "      2x        2x(1 + x2)   \r\n" );
document.write( "   ————————  = ————————————\r\n" );
document.write( "    1 - x2        1 - x4\r\n" );
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document.write( "Factor the denominator on the right as the\r\n" );
document.write( "difference of two perfect squares:\r\n" );
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document.write( "      2x           2x(1 + x2)   \r\n" );
document.write( "   ————————  = ——————————————————\r\n" );
document.write( "    1 - x2      (1 - x2)(1 + x2)\r\n" );
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document.write( "Cancel the (1 + x2)'s on the right:\r\n" );
document.write( "                        1\r\n" );
document.write( "      2x           2x(1 + x2)   \r\n" );
document.write( "   ————————  = ——————————————————\r\n" );
document.write( "    1 - x2      (1 - x2)(1 + x2)\r\n" );
document.write( "                           1\r\n" );
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document.write( "      2x           2x   \r\n" );
document.write( "   ————————  = ——————————\r\n" );
document.write( "    1 - x2       1 - x2\r\n" );
document.write( "                           \r\n" );
document.write( "Edwin

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