document.write( "Question 1068643: cos^3(A)×cos(3A)+sin^3(A)×sin(3A)=cos^3 (2A) \n" ); document.write( "

Algebra.Com's Answer #683916 by ikleyn(52788) $\"\"$ $\"About$

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\n" ); document.write( "cos^3(A)×cos(3A)+sin^3(A)×sin(3A)=cos^3 (2A)
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\n" ); document.write( "\n" ); document.write( "Unfortunately, the person who submitted this post didn't say what he/she wanted.\r
\n" ); document.write( "\n" ); document.write( "So, I will formulate it instead of him/her: \"Prove an identity . . . \"\r
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document.write( "The keys are these two formulas:\r\n" );
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document.write( "      = ,        (1)   and\r\n" );
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document.write( "      = .        (2)\r\n" );
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document.write( "(see the lesson Powers of trigonometric functions in this site).  When applying them, you will get\r\n" );
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document.write( "cos^3(A)*cos(3A) =  ,   (3)  and\r\n" );
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document.write( "sin^3(A)*sin(3A) = .   (4)\r\n" );
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document.write( "So, adding and expanding (3) and (4), you will get\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( "    For the first  bracket  [ . . ]  apply the formula cos(2x) = . . .   \r\n" );
document.write( "    For the second bracket  [ . . ]  apply the formula cos(x-y) = . . .   You will get\r\n" );
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document.write( "=  =  = \r\n" );
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document.write( "    Now apply again the formula (1). You will get\r\n" );
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document.write( "= .\r\n" );
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\n" ); document.write( "\n" ); document.write( "QED. Proved and solved.\r
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