document.write( "Question 1165987: Use this identify to solve the question:\r
\n" ); document.write( "\n" ); document.write( "tan4Q = $\"%284tanQ-4%28tanQ%29%5E3%29%2F%281-6%28tanQ%29%5E2%2B%28tanQ%29%5E4%29\"$ \r
\n" ); document.write( "\n" ); document.write( "Find the polynomial of least degree that has zeroes $\"%28cot%28pi%2F24%29%29%5E2\"$ , $\"%28cot%287pi%2F24%29%29%5E2\"$ , $\"%28cot%2813pi%2F24%29%29%5E2\"$ , $\"%28cot%2819pi%2F24%29%29%5E2%29\"$ \r
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Algebra.Com's Answer #790469 by ikleyn(52908) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "Use this identify to solve the question:\r
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\n" ); document.write( "\n" ); document.write( "tan4Q = $\"%284tanQ-4%28tanQ%29%5E3%29%2F%281-6%28tanQ%29%5E2%2B%28tanQ%29%5E4%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Find the polynomial of least degree that has zeroes $\"%28cot%28pi%2F24%29%29%5E2\"$ , $\"%28cot%287pi%2F24%29%29%5E2\"$ , $\"%28cot%2813pi%2F24%29%29%5E2\"$ , $\"%28cot%2819pi%2F24%29%29%5E2%29\"$
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\n" ); document.write( "\n" ); document.write( "So, according to the problem formulation, I should accept the given identity \"as is\", and based on it,
\n" ); document.write( "to construct the polynomial.\r
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\n" ); document.write( "\n" ); document.write( "Since this problem is designed/intendent for advanced students, I will only show the major idea and the major steps
\n" ); document.write( "without going in details.\r
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\n" ); document.write( "\n" ); document.write( "Let me start noticing that of four numbers $\"%28cot%28pi%2F24%29%29%5E2\"$ , $\"%28cot%287pi%2F24%29%29%5E2\"$ , $\"%28cot%2813pi%2F24%29%29%5E2\"$ , $\"%28cot%2819pi%2F24%29%29%5E2%29\"$ , \r
\n" ); document.write( "\n" ); document.write( "the first is equal to the third, while the second is equal to the fourth.\r
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\n" ); document.write( "\n" ); document.write( "So, I will construct the polynomial which has two zeroes as the first and the second numbers of the four numbers listed:
\n" ); document.write( "then the third and the fourth numbers will be the roots of this polynomial automatically . . . \r
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document.write( "Take Q = .  Then  4Q =   and  tan(4Q) =  = .\r\n" );
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document.write( "According to the given identity, I have then\r\n" );
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document.write( "     =   or\r\n" );
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document.write( "     = .    (2)\r\n" );
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document.write( "Next, in (2), I will replace (tan(Q))^2  by x everywhere (at each appearance).\r\n" );
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document.write( "I will get then\r\n" );
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document.write( "     = .    (3)\r\n" );
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document.write( "Now, for   there is the formula (the known expression)  \r\n" );
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document.write( "          =   (see the link  https://brainly.in/question/9263846 )\r\n" );
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document.write( "and/or   \r\n" );
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document.write( "          =   (see the link  https://mathworld.wolfram.com/TrigonometryAnglesPi24.html)\r\n" );
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document.write( "Replacing    in  formula  (3)  by any of these constant expressions (they are equal (!)),\r\n" );
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document.write( "I get the polynomial of the degree 2\r\n" );
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document.write( "     = ,           (4)\r\n" );
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document.write( "whose two roots are    and  .\r\n" );
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document.write( "If you want to have the polynomial with the roots    and  ,\r\n" );
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document.write( "simply replace x in the polynomial (4) by   and transform the obtained rational function to the form\r\n" );
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document.write( "of the ratio    with polynomials  P(x) and Q(x).\r\n" );
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document.write( "Then the polynomial  P(x)  will be the answer to the problem's question.\r\n" );
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document.write( "Probably, it is more correctly in English to say \"a polynomial P(x)\", because every product of such a polynomial by a constant\r\n" );
document.write( "satisfies the problem's conditions, too.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved and explained.\r
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\n" ); document.write( "\n" ); document.write( "In such problems, knowing the way is more important than gettting the exact answer.\r
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