document.write( "Question 1093836: Solve the equation in [0,2π] by squaring both sides.
\n" ); document.write( "Sec x-tanx=-1
\n" ); document.write( "Select one:
\n" ); document.write( " a. 3π
\n" ); document.write( " b. 4π
\n" ); document.write( " c. 2π
\n" ); document.write( " d. π \n" ); document.write( "

Algebra.Com's Answer #708456 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
If the answer must be between $\"0\"$ and $\"2pi\"$ ,
\n" ); document.write( " $\"3pi\"$ and $\"4pi\"$ are obviously wrong answers.
\n" ); document.write( " $\"sec%28x%29=1%2Fcos%28x%29\"$
\n" ); document.write( "For $\"x=pi\"$ , $\"tan%28x%29=tan%28pi%29=0\"$ and $\"cos%28x%29=cos%28pi%29=-1\"$ ,
\n" ); document.write( "so $\"sec%28x%29=1%2F%28-1%29=-1\"$ ,
\n" ); document.write( "and $\"sec%28x%29-tan%28x%29=-1-0=-1\"$ , so $\"x=highlight%28pi%29\"$ is a solution.
\n" ); document.write( "For $\"x=2pi\"$ , $\"tan%28x%29=0\"$ too, but $\"cos%28x%29=1\"$ ,
\n" ); document.write( "so $\"sec%28x%29-tan%28x%29r1-0=1\"$ , and $\"x=2pi\"$ is not a solution.
\n" ); document.write( "
\n" ); document.write( "IF YOU HAD TO \"SHOW YOUR WORK\" you would have to solve the equation,
\n" ); document.write( "and for that squaring is a reasonable suggestion.
\n" ); document.write( " $\"sec%28x%29-tan%28x%29=-1\"$
\n" ); document.write( " $\"1%2Fcos%28x%29-sin%28x%29%2Fcos%28x%29=-1\"$
\n" ); document.write( "Obviously $\"cos%28x%29=0\"$ is not allowed.
\n" ); document.write( "Multiplying times $\"cos%28x%29\"$ ,
\n" ); document.write( "as long as $\"cos%28x%29%3C%3E0\"$ ,
\n" ); document.write( "gives an equivalent equation.
\n" ); document.write( " $\"1-sin%28x%29=-cos%28x%29\"$
\n" ); document.write( "Now you can square,
\n" ); document.write( "but the resulting equation will have
\n" ); document.write( "all the solutions to $\"1-sin%28x%29=-cos%28x%29\"$ , and
\n" ); document.write( "all the solutions to $\"1-sin%28x%29=cos+%28x%29\"$ ,
\n" ); document.write( "so you will have to check for and discard any extraneous solutions.
\n" ); document.write( "Squaring, you get
\n" ); document.write( " $\"1%2Bsin%5E2%28x%29-2sin%28x%29=cos%5E2%28x%29\"$ ,
\n" ); document.write( "which can be re-written as
\n" ); document.write( " $\"1%2Bsin%5E2%28x%29-wish%28x%29=1-sin%5E2%28x%29\"$ .
\n" ); document.write( "Simplifying and rearranging,
\n" ); document.write( " $\"sin%5E2%28x%29-2sin%28x%29=-sin%5E2%28x%29\"$
\n" ); document.write( " $\"2sin%5E2%28x%29-2sin%28x%29=0\"$ .
\n" ); document.write( "Factoring, gives you
\n" ); document.write( " $\"2sin%28x%29%281-sin%28x%29%29=0\"$
\n" ); document.write( "The solutions need to have
\n" ); document.write( " $\"sin%28x%29=0\"$ or $\"sin%28x%29=1\"$ .
\n" ); document.write( "However, $\"sin%281%29=0\"$ means $\"cos%28x%29=0\"$ ,
\n" ); document.write( "and as said at the beginning, that is not allowed.
\n" ); document.write( "We are left with $\"sin%28x%29=0\"$ ,
\n" ); document.write( "which means $\"system%28cos%28x%29=1%2C%22or%22%2Ccos%28x%29=-1%29\"$ .
\n" ); document.write( "Checking for extraneous solutions,
\n" ); document.write( "we would see that
\n" ); document.write( " $\"system%28sin%28x%29=0%2Ccos%28x%29=1%29\"$
\n" ); document.write( "is a solution of
\n" ); document.write( " $\"1-sin%28x%29=cos%28x%29\"$ ,
\n" ); document.write( "and is an extraneous solution.
\n" ); document.write( "That leaves us with $\"cos%28x%29=-1\"$ ,
\n" ); document.write( "which in the interval given happens only when
\n" ); document.write( " $\"highlight%28x=pi%29\"$ . \n" ); document.write( "

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