document.write( "Question 1025159: Im stuck. I need the exact solution from [o.2pi)\r
\n" ); document.write( "\n" ); document.write( "tan^2(x)= 1-sec(x) \n" ); document.write( "

Algebra.Com's Answer #640517 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
Using the trigonometric identities that we should all remember,
\n" ); document.write( " $\"tan%28x%29=sin%28x%29%2Fcos%28x%29\"$ and $\"sec%28x%29=1%2Fcos%28x%29\"$ ,
\n" ); document.write( "you can re-write the equation as
\n" ); document.write( " $\"sin%5E2%28x%29%2Fcos%5E2%28x%29=1-1%2Fcos%28x%29\"$ .
\n" ); document.write( "Multiplying both sides of the equal sign times $\"cos%5E2%28x%29\"$ ,
\n" ); document.write( "the equation above turns into
\n" ); document.write( " $\"sin%5E2%28x%29=cos%5E2%28x%29-cos%28x%29\"$ ,
\n" ); document.write( "and since $\"sin%5E2%28x%29=1-cos%5E2%28x%29\"$ ,
\n" ); document.write( "you can substitute and re-write it as
\n" ); document.write( " $\"1-cos%5E2%28x%29%7D=cos%5E2%28x%29-cos%28x%29\"$ <--> $\"0=2cos%5E2%28x%29-cos%28x%29-1\"$ .
\n" ); document.write( "The last equation is a quadratic equation.
\n" ); document.write( "If you use $\"y=cos%28x%29\"$ , you can write it as
\n" ); document.write( " $\"2y%5E2-y-1=0\"$
\n" ); document.write( "You can solve for $\"y\"$ using factoring, or completing the square, or the quadratic formula to get the solutions
\n" ); document.write( " $\"system%28y=1%2C%22or%22%2Cy=-1%2F2%29\"$ .
\n" ); document.write( "In the interval $\"%22%5B+0+%2C%22\"$ $\"2pi\"$ $\"%22%29%22\"$ ,
\n" ); document.write( " $\"cos%28x%29=1\"$ happens only for $\"highlight%28x=0%29\"$ ,
\n" ); document.write( "while $\"cos%28x%29=-1%2F2\"$ happens only for $\"highlight%28system%28x=2pi%2F3%2C%22or%22%2Cx=4pi%2F3%29%29\"$ .
\n" ); document.write( " \n" ); document.write( "

\n" );