document.write( "Question 1199347: If tan(x/2)=sinx and cosx≠0, then the value of tanx is… \n" ); document.write( "

Algebra.Com's Answer #833196 by ikleyn(52803) $\"\"$ $\"About$

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\n" ); document.write( "If tan(x/2)=sin(x) and cos(x)≠0, then the value of tan(x) is . . .
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document.write( "Our starting equation is \r\n" );
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document.write( "     = .      (1)\r\n" );
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document.write( "Since  is not defined when  = 0, the domain of this equation is \r\n" );
document.write( "the set of all real numbers except  = ,  or  x =  = .\r\n" );
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document.write( "One obvious solution to equation (1) is x = , when both sides of equation (1) are zeroes.    (2)\r\n" );
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document.write( "In what follows, we assume that x is neither  nor .    (3)\r\n" );
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document.write( "        In particular, it implies that  =/= 0.    (4)\r\n" );
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document.write( "Now we transform equation (1) step by step\r\n" );
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document.write( "     = \r\n" );
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document.write( "     = \r\n" );
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document.write( "Due to (4), we can divide both sides by , since this factor is not zero.\r\n" );
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document.write( "     = \r\n" );
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document.write( "     = \r\n" );
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document.write( "     = \r\n" );
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document.write( "     =  = \r\n" );
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document.write( "     =     <<<---=== notice here the difference with the Edwin's solution\r\n" );
document.write( "                                 (here Edwin lost some/(half of the) solutions)\r\n" );
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document.write( "     =  =  = , where n is any integer.\r\n" );
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document.write( "              But for all these values of x, cos(x) = 0, so we should exclude this set of values\r\n" );
document.write( "              from our consideration due to condition cos(x) =/= 0, imposed in the problem.\r\n" );
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document.write( "Finally, the only set of solutions is  x = , found in (2).\r\n" );
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document.write( "ANSWER.  The solution to equation (1) with the imposed condition cos(x) =/= 0 \r\n" );
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document.write( "         is the set  x = , and tan(x) = 0 for all these values of x.\r\n" );
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