document.write( "Question 1004522: Prove that the equation is an identity\r
\n" ); document.write( "\n" ); document.write( " $\"+%28sin+3x+%2B+sin+5x%29+%2F+%28sin+3x+-+sin+5x%29+=+-+%28tan+4x%29+%2F+tan+x+\"$ \n" ); document.write( "

Algebra.Com's Answer #620985 by ikleyn(52847) $\"\"$ $\"About$

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\n" ); document.write( "Apply the formulas for addition and subtraction of trigonometric functions (see the lesson \r
\n" ); document.write( "\n" ); document.write( "Addition and subtraction of trigonometric functions in this site) to the numerator and denominator.\r
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\n" ); document.write( "\n" ); document.write( "Let start with the numerator.\r
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\n" ); document.write( "\n" ); document.write( "sin(3x) + sin(5x) = 2*sin( $\"%283x%2B5x%29%2F2\"$ )*cos( $\"%283x-5x%29%2F2\"$ ) = 2*sin(4x)*cos(-x) = 2*sin4x)*cos(x). <----- Recall that cos(-x) = cos(x).\r
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\n" ); document.write( "\n" ); document.write( "Similarly for the denominator\r
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\n" ); document.write( "\n" ); document.write( "sin(3x) - sin(5x) = 2*sin( $\"%283x-5x%29%2F2\"$ )*cos( $\"%283x%2B5x%29%2F2\"$ ) = 2*sin(-x)*cos(4x) = -2*sin(x)*cos(4x). <----- Recall that sin(-x) = -sin(x).\r
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\n" ); document.write( "\n" ); document.write( "Thus \r
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\n" ); document.write( "\n" ); document.write( " $\"%28sin+3x+%2B+sin+5x%29%2F%28sin+3x+-+sin+5x%29\"$ = $\"-%282%2Asin%284x%29%2Acos%28x%29%29%2F%282%2Asin%28x%29%2Acos%284x%29%29\"$ = - $\"%28sin%284x%29%2Fcos%284x%29%29%2A%28cos%28x%29%2Fsin%28x%29%29\"$ = - $\"%28tan%284x%29%29%2F%28tan%28x%29%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "It is exactly what has to be proved.\r
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