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document.write( "Hi, there--\r\n" );
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document.write( "THE PROBLEM:\r\n" );
document.write( "I am given the expression:\r\n" );
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document.write( "Which of the following expressions is the above expression NOT equal to?\r\n" );
document.write( "A. 
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document.write( "B. 
\r\n" );
document.write( "C. 
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document.write( "D. 1\r\n" );
document.write( "E. 
\r\n" );
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document.write( "I *think* I've narrowed it down to either A, C or E, but am not sure. Thank you for your help!\r\n" );
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document.write( "A SOLUTION:\r\n" );
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document.write( "SOME IDEAS\r\n" );
document.write( "One strategy is to rewrite the expressions in terms of sin(x) and cos(x). This can make it easier to see \r\n" );
document.write( "what you have going on. Let's try that with .\r\n" );
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document.write( "Now simplify. Change division by sin(x)/cos(x) to multiplication by its reciprocal.\r\n" );
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document.write( "Now we can cancel out sin(x) and cos(x) in the numerator and denominator.\r\n" );
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document.write( "We see that \r\n" );
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document.write( "Look! We have already eliminated answer [D].\r\n" );
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document.write( "Let's try this with answer [A].\r\n" );
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document.write( "Rewrite the right-hand side as one fraction with a denominator of sin^2(x).\r\n" );
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document.write( "A trig identity that comes in handy ALL the time is sin^2(x) + cos^2(x) = 1. We can rewrite this identity\r\n" );
document.write( "as sin^2(x) = 1 - cos^2(x).\r\n" );
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document.write( "Substitute sin^2(x) for 1 - cos^2(x) in the numerator.\r\n" );
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document.write( "We see that this expression is also equal to 1. We have eliminated answer [A].\r\n" );
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document.write( "You can try this technique with answer [C]. \r\n" );
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document.write( "For answer [E]: \r\n" );
document.write( "We know that our original expression is equal to 1. Recall that cos^2(x) + sin^2(x) = 1. \r\n" );
document.write( "If we can show that cos^(2)(x) - sin^(2)(x) is not equal cos^(x) + sin^2(x), we will know that answer \r\n" );
document.write( "[E] is not equivalent to the original expression. \r\n" );
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document.write( "Suppose that 
.\r\n" );
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document.write( "Subtract cos^2(x) from both sides.\r\n" );
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document.write( "Subtract sin^2(x) from both sides of the equation.\r\n" );
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document.write( "Divide both sides by -2.\r\n" );
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document.write( "Take the square root of both sides.\r\n" );
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document.write( "This tells us that 
and 
are only equal when sin(x) = 0.\r\n" );
document.write( "Therefore, answer [E] is not equivalent to the original expression for all values of x.\r\n" );
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document.write( "That leaves answer [B] to check. If you graph the expression, you will see that it is not equal to 1 for all values of x. \r\n" );
document.write( "However, proving that algebraically is a little messy. If you are interested in seeing the math. Email me and I'll add it here.\r\n" );
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document.write( "Hope this helps! Feel free to email if you have ANY questions about the solution.\r\n" );
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document.write( "Good luck with your math,\r\n" );
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document.write( "Mrs. F\r\n" );
document.write( "math.in.the.vortex@gmail.com\r\n" );
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