document.write( "Question 254929: find all solutions:\r
\n" ); document.write( "\n" ); document.write( "cscx + cotx = 1 ; in the interval [0,2pi) \n" ); document.write( "

Algebra.Com's Answer #187228 by drk(1908) $\"\"$ $\"About$

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find all solutions:
\n" ); document.write( " $\"csc%28x%29+%2B+cot%28x%29+=+1\"$ ; in the interval [0,2pi)
\n" ); document.write( "first
\n" ); document.write( "csc(x) = 1/sin(x)
\n" ); document.write( "cot(x) = cos(x) / sin(x)
\n" ); document.write( "so we get
\n" ); document.write( " $\"1%2Fsin%28x%29+%2B+cos%28x%29+%2F+sin%28x%29+=+1\"$
\n" ); document.write( "multiply by sin(x) to get
\n" ); document.write( " $\"1+%2B+cos%28x%29+=+sin%28x%29\"$
\n" ); document.write( " $\"sin%5E2%28x%29+=+1+-+cos%5E2%28x%29+=+%281%2Bcos%28x%29%29%281-cos%28x%29%29\"$
\n" ); document.write( "substituting, we get
\n" ); document.write( " $\"1+%2B+cos%28x%29+=+sqrt%281%2Bcos%28x%29%29%281-cos%28x%29%29\"$
\n" ); document.write( "squaring boh sides, we get
\n" ); document.write( " $\"%281+%2B+cos%28x%29%29%281%2Bcos%28x%29%29+=+%281%2Bcos%28x%29%29%281-cos%28x%29%29\"$
\n" ); document.write( "dividing by (1+cos(x)), we get
\n" ); document.write( " $\"1+%2B+cos%28x%29+=+1-cos%28x%29\"$
\n" ); document.write( " $\"cos%28x%29+=+0\"$
\n" ); document.write( " $\"x+=+pi%2F2\"$
\n" ); document.write( " $\"x+=+3pi%2F2\"$
\n" ); document.write( "--
\n" ); document.write( "If x = pi/2,
\n" ); document.write( "then
\n" ); document.write( "csc(x) = 1 and cot(x) = 0; 1 + 0 = 1
\n" ); document.write( "if x = 3pi/2,
\n" ); document.write( "then
\n" ); document.write( "csc(x) = -1 and cot(x) = 0 ; -1 + 0 not = 1.
\n" ); document.write( "So it appear that our only answer is
\n" ); document.write( "X = PI/2. \n" ); document.write( "

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