document.write( "Question 1208297: Simplify
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Algebra.Com's Answer #846631 by Edwin McCravy(20054)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "\"%281%2F%28tan%28x%29+%2B+cot%28x%29+%2B+sec%28x%29+%2B+csc%28x%29%29%29%5E2\" \r\n" );
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document.write( "\"1%5E%22%22%2F%28tan%28x%29+%2B+cot%28x%29+%2B+sec%28x%29+%2B+csc%28x%29%5E%22%22%29%5E2\"\r\n" );
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document.write( "Draw a right triangle with angle x, hypotenuse 1, and legs \"a\" and \"b\"\r\n" );
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document.write( "\"1%5E%22%22%2F%28a%2Fb+%2B+b%2Fa+%2B+1%2Fb+%2B+1%2Fa%29%5E2\"\r\n" );
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document.write( "Since , that's\r\n" );
document.write( "because by the Pythagorean theorem, a2+b2=1\r\n" );
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document.write( "\"1%5E%22%22%2F%28%281%2Ba%2Bb%29%2F%28ab%29%29%5E2\"\r\n" );
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document.write( "Multiply top and bottom by a2b2\r\n" );
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document.write( "\"a%5E2b%5E2%2F%281%2Ba%2Bb%29%5E2\"\r\n" );
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document.write( "We multiply out the denominator,\r\n" );
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document.write( "\"a%5E2b%5E2%2F%281%2Ba%5E2%2Bb%5E2%2B2a%2B2b%2B2ab%29\"\r\n" );
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document.write( "Since a2+b2=1,\r\n" );
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document.write( "\"a%5E2b%5E2%2F%282%2B2a%2B2b%2B2ab%29\"\r\n" );
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document.write( "\"expr%281%5E%22%22%2F2%5E%22%22%29%2A%28a%5E2b%5E2%2F%281%2Ba%2Bb%2Bab%5E%22%22%29%29\"\r\n" );
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document.write( "The denominator factors\r\n" );
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document.write( "\"expr%281%5E%22%22%2F2%5E%22%22%29%2A%28a%5E2b%5E2%2F%28%281%2Ba%29%281%2Bb%29%29%29\"\r\n" );
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document.write( "Multiply by the product of the \"conjugates\" of the\r\n" );
document.write( "factors on the bottom over itself. (Sort of like\r\n" );
document.write( "rationalizing the denominator)\r\n" );
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document.write( "\"expr%281%5E%22%22%2F2%5E%22%22%29%2A%28a%5E2b%5E2%2F%28%281%2Ba%29%281%2Bb%29%29%29\"\"%22%22%2A%22%22\"\"%28%28%281-a%29%281-b%29%5E%22%22%29%2F%28%281-a%29%281-b%29%5E%22%22%29%29\"\r\n" );
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document.write( " \r\n" );
document.write( "\"expr%281%2F2%29%2A%28a%5E2b%5E2%281-a%29%281-b%29%29%2F%28%281-a%5E2%29%281-b%5E2%29%29%29\"\r\n" );
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document.write( "By the Pythagorean theorem  1-a2=b2 and 1-b2=a2\r\n" );
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document.write( "\"expr%281%2F2%29%2A%28a%5E2b%5E2%281-a%29%281-b%29%29%2F%28b%5E2a%5E2%29%29%29\"\r\n" );
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document.write( "Canceling:\r\n" );
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document.write( "\"expr%281%2F2%29%2A%281-a%29%281-b%29\"\r\n" );
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document.write( "Since a = sin(x) and b = cos(x)\r\n" );
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document.write( "\"expr%281%2F2%29%2A%281%5E%22%22-sin%28x%29%29%281%5E%22%22-cos%28x%29%29\"  <--simplest form\r\n" );
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document.write( "Edwin
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