document.write( "Question 1116384: Use the complex number cosx + isinx to find the constants A, B, C in the identity
\n" ); document.write( "cos4x = Acos^4x + Bcos^2xSin^2x + Csin^4x
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Algebra.Com's Answer #731277 by ikleyn(52776)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "z = cos(x) + i*sin(x)  ====>\r\n" );
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document.write( "\"z%5E4\" = \"%28cos%28x%29+%2B+i%2Asin%28x%29%29%5E4\" = de Moivre = cos(4x) + i*sin(4x)  ====>\r\n" );
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document.write( "cos^4(x) + 4*i*cos^3(x)*sin(x) + 6*i^2*cos^2(x)*sin^2(x) + 4*i^3*cos(x)*sin^3(x) + i^4*sin^4(x) = cos(4x) + i*sin(4x) ,   or, equivalently\r\n" );
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document.write( "cos^4(x) + 4*i*cos^3(x)*sin(x) - 6*cos^2(x)*sin^2(x) - 4*i*cos(x)*sin^3(x) + sin^4(x) = cos(4x) + i*sin(4x),   which implies\r\n" );
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document.write( "cos(4x) = \"cos%5E4%28x%29+-+6%2Acos%5E2%28x%29%2Asin%5E2%28x%29++%2B+sin%5E4%28x%29\"\r\n" );
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document.write( "which is an identity you are looking for.\r\n" );
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document.write( "So, A = 1,  B = -6,  C = 1.\r\n" );
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\n" ); document.write( "On complex numbers, see the lessons \r
\n" ); document.write( "\n" ); document.write( "    - Complex numbers and arithmetical operations on them\r
\n" ); document.write( "\n" ); document.write( "    - Complex plane\r
\n" ); document.write( "\n" ); document.write( "    - Addition and subtraction of complex numbers in complex plane\r
\n" ); document.write( "\n" ); document.write( "    - Multiplication and division of complex numbers in complex plane\r
\n" ); document.write( "\n" ); document.write( "    - Raising a complex number to an integer power\r
\n" ); document.write( "\n" ); document.write( "    - How to take a root of a complex number\r
\n" ); document.write( "\n" ); document.write( "    - Solution of the quadratic equation with real coefficients on complex domain\r
\n" ); document.write( "\n" ); document.write( "    - How to take a square root of a complex number\r
\n" ); document.write( "\n" ); document.write( "    - Solution of the quadratic equation with complex coefficients on complex domain\r
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\n" ); document.write( "\n" ); document.write( "    - Solved problems on taking roots of complex numbers\r
\n" ); document.write( "\n" ); document.write( "    - Solved problems on arithmetic operations on complex numbers\r
\n" ); document.write( "\n" ); document.write( "    - Solved problem on taking square root of complex number\r
\n" ); document.write( "\n" ); document.write( "    - Miscellaneous problems on complex numbers\r
\n" ); document.write( "\n" ); document.write( "    - Advanced problem on complex numbers\r
\n" ); document.write( "\n" ); document.write( "    - Solved problems on de'Moivre formula\r
\n" ); document.write( "\n" ); document.write( "    - A curious example of an equation in complex numbers which HAS NO a solution\r
\n" ); document.write( "\n" ); document.write( "in this site.\r
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\n" ); document.write( "\n" ); document.write( "Also,  you have this free of charge online textbook in ALGEBRA-II in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this online textbook under the topic  \"Complex numbers\".\r
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\n" ); document.write( "\n" ); document.write( "Save the link to this textbook together with its description\r
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\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-II
\n" ); document.write( "https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson\r
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\n" ); document.write( "\n" ); document.write( "into your archive and use when it is needed.\r
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