document.write( "Question 276355: Hi. I have a question about Pythagorean identities. I do not understrand how to write the espression as an integer. The problem is:\r
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Algebra.Com's Answer #201429 by Edwin McCravy(20065)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "First let's learn where the Pythagorean identities came from,\r\n" );
document.write( "namely the Pythagorean theorem:\r\n" );
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document.write( "By the Pythagorean theorem:\r\n" );
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document.write( "\"adjacent%5E2+%2B+opposite%5E2+=+hypotenuse%5E2\"\r\n" );
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document.write( "Divide through by \"hypotenuse%5E2\"\r\n" );
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document.write( "\"Cos%5E2theta+%2B+Sin%5E2theta+=+1\"\r\n" );
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document.write( "That's the first Pythagorean identity:\r\n" );
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document.write( "\"adjacent%5E2+%2B+opposite%5E2+=+hypotenuse%5E2\"\r\n" );
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document.write( "Divide through by \"adjacent%5E2\"\r\n" );
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document.write( "\"1+%2B+%28%28opposite%29%2F%28hypotenuse%29%29%5E2+=+%28%28hypotenuse%29%2Fadjacent%29%29%5E2\"\r\n" );
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document.write( "\"1+%2B+Tan%5E2theta+=+Sec%5E2theta\"\r\n" );
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document.write( "That's the second Pythagorean identity:\r\n" );
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document.write( "\"adjacent%5E2+%2B+opposite%5E2+=+hypotenuse%5E2\"\r\n" );
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document.write( "Divide through by \"opposite%5E2\"\r\n" );
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document.write( "\"Cot%5E2theta+%2B+1+=+Csc%5E2theta\"\r\n" );
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document.write( "That's the third Pythagorean identity:\r\n" );
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document.write( "Be sure to memorize all three of them.  Usually we write them\r\n" );
document.write( "this way, so they're easier to learn:\r\n" );
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document.write( "\"Sin%5E2theta+%2B+Cos%5E2theta+=+1\"\r\n" );
document.write( "\"1+%2B+Tan%5E2theta+=+Sec%5E2theta\"\r\n" );
document.write( "\"1+%2B+Cot%5E2theta+=+Csc%5E2theta\"\r\n" );
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document.write( "Those identities are true for ALL angles \"theta\"\r\n" );
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document.write( "So for\r\n" );
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document.write( "\"Csc%5E2\"\"%283alpha%29\"\"%22%22-%22%22\"\"Cot%5E2\"\"%283alpha%29\"\r\n" );
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document.write( "Since the third Pythagorean identity\r\n" );
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document.write( "\"1+%2B+Cot%5E2theta+=+Csc%5E2theta\" \r\n" );
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document.write( "is true for ALL angles \"theta\", let \"theta+=+3alpha\", then\r\n" );
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document.write( "\"1+%2B+Cot%5E2\"\"%283alpha%29\"\"%22%22=%22%22\"\"Csc%5E2\"\"%283alpha%29\"\r\n" );
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document.write( "So we can replace \"Csc%5E2\"\"%283alpha%29\" by \"1+%2B+Cot%5E2\"\"%283alpha%29\"\r\n" );
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document.write( "So your problem:\r\n" );
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document.write( "\"Csc%5E2\"\"%283alpha%29\"\"%22%22-%22%22\"\"Cot%5E2\"\"%283alpha%29\"\r\n" );
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document.write( "becomes:\r\n" );
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document.write( "\"1+%2B+Cot%5E2\"\"%283alpha%29\"\"%22%22-%22%22\"\"Cot%5E2\"\"%283alpha%29\"\r\n" );
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document.write( "then the \"cotangent squared\"s cancel:\r\n" );
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document.write( "\"1+%2B+cross%28Cot%5E2%29\"\"%28cross%283alpha%29%29\"\"%22%22-%22%22\"\"cross%28Cot%5E2%29\"\"%28cross%283alpha%29%29\"\r\n" );
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document.write( "and that just leaves 1, which is an integer.\r\n" );
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document.write( "Edwin
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