document.write( "Question 515391: Simplify do not solve (cos x)/(csc x - sin x) \n" ); document.write( "

Algebra.Com's Answer #343949 by drcole(72) $\"\"$ $\"About$

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Recall that $\"csc+x+=+%281%29%2F%28sin+x%29\"$ . So we substitute this into our expression and work from there:\r
\n" ); document.write( "\n" ); document.write( " $\"+%28cos+x%29%2F%28csc+x+-+sin+x%29+\"$
\n" ); document.write( " $\"+%28cos+x%29%2F%28%281%29%2F%28sin+x%29+-+sin+x%29%29\"$ (substituting)
\n" ); document.write( " $\"+%28cos+x%29%2F%28%281%29%2F%28sin+x%29+-+%28sin+x%29%5E2%2F%28sin+x%29%29\"$ (putting all of the terms in the denominator under a common denominator)
\n" ); document.write( " $\"+%28cos+x%29%2F%28%281+-+%28sin+x%29%5E2%29%2F%28sin+x%29%29\"$ (combining fractions in the denominator)
\n" ); document.write( " $\"+%28cos+x%29+%2A+%28%28sin+x%29%2F%281+-+%28sin+x%29%5E2%29%29\"$ (simplifying the complex fraction by multiplying by the reciprocal of the denominator)\r
\n" ); document.write( "\n" ); document.write( "Now recall that $\"%28cos+x%29%5E2+%2B+%28sin+x%29%5E2+=+1\"$ , so $\"%28cos+x%29%5E2+=+1+-+%28sin+x%29%5E2\"$ . So, substituting $\"%28cos+x%29%5E2\"$ for $\"1+-+%28sin+x%29%5E2\"$ , we get:\r
\n" ); document.write( "\n" ); document.write( " $\"+%28cos+x%29+%2A+%28%28sin+x%29%2F%28%28cos+x%29%5E2%29%29\"$ (substituting)
\n" ); document.write( " $\"+1+%2A+%28%28sin+x%29%2F%28cos+x%29%29\"$ (canceling one $\"cos+x\"$ )
\n" ); document.write( " $\"+tan+x+\"$ (remembering that $\"tan+x+=+%28sin+x%29%2F%28cos+x%29\"$ )\r
\n" ); document.write( "\n" ); document.write( "So our expression simplified to $\"tan+x\"$ . \n" ); document.write( "

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