document.write( "Question 376402: hi could you please help me to verify this identity
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\n" ); document.write( "(cot^3 x - tan^3 x)/(sec^2x + cot^2x) = 2cot(2x) \n" ); document.write( "
Algebra.Com's Answer #267820 by jsmallt9(3758)\"\" \"About 
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\n" ); document.write( "First, we can use the pattern, \"a%5E3-b%5E3+=+%28a-b%29%28a%5E2%2Bab%2Bb%5E2%29\", to factor the numerator:
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\n" ); document.write( "Since cot and tan are reciprocals, cot(x)*tan(x) = 1:
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\n" ); document.write( "Since \"1+%2B+tas%5E2%28x%29+=+sec%5E2%28x%29\":
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\n" ); document.write( "We can now see that the fraction will reduce:
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\n" ); document.write( "leaving:
\n" ); document.write( "\"cot%28x%29+-+tan%28x%29+=+2cot%282x%29\"
\n" ); document.write( "Rewriting the left side using sin and cos (a common technique use in these problems) we get:
\n" ); document.write( "\"cos%28x%29%2Fsin%28x%29+-+sin%28x%29%2Fcos%28x%29+=+2cot%282x%29\"
\n" ); document.write( "Since the right side has one term, we probably want one term on the left side, too. So we will add the fractions. Of course we need common denominators first:
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\n" ); document.write( "Now we can subtract:
\n" ); document.write( "\"%28cos%5E2%28x%29-sin%5E2%28x%29%29%2F%28sin%28x%29cos%28x%29%29+=+2cot%282x%29\"
\n" ); document.write( "The numerator exactly matches the formula for cos(2x). The denominator is close to sin(2x). sin(2x) = 2sin(x)cos(x). If we multiply both sides by 1/2 we get: \"%281%2F2%29sin%282x%29+=+sin%28x%29cos%28x%29\". Substituting these into our fraction we get:
\n" ); document.write( "\"cos%282x%29%2F%28%281%2F2%29sin%28x%29cos%28x%29%29+=+2cot%282x%29\"
\n" ); document.write( "We can get rid of the fraction in the denominator by multiplying the numerator and denominator by 2:
\n" ); document.write( "\"%28cos%282x%29%2F%28%281%2F2%29sin%282x%29%29%29%282%2F2%29+=+2cot%282x%29\"
\n" ); document.write( "giving:
\n" ); document.write( "\"%282cos%282x%29%29%2Fsin%282x%29+=+2cot%282x%29\"
\n" ); document.write( "or
\n" ); document.write( "\"2%28cos%282x%29%2Fsin%282x%29%29+=+2cot%282x%29\"
\n" ); document.write( "The fraction is cot(2x):
\n" ); document.write( "2cot(2x) = 2cot(2x) Done! \n" ); document.write( "
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