document.write( "Question 969538: 1-sinx/cosx=cosx/1+sinx \n" ); document.write( "
Algebra.Com's Answer #592484 by KMST(5328)\"\" \"About 
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NOTE:
\n" ); document.write( "I believe you meant \"%281-sin%28x%29%29%2Fcos%28x%29=%281-sin%28x%29%29%2Fcos%28x%29%29\" .
\n" ); document.write( "If that was not what you meant, then I solved the wrong problem.
\n" ); document.write( "
\n" ); document.write( "You could write \"%281-sin%28x%29%29%2Fcos%28x%29=%281-sin%28x%29%29%2Fcos%28x%29%29\" as
\n" ); document.write( "(1-sinx)/cosx=cosx/(1+sinx) , because
\n" ); document.write( "1-sin(x)/cos(x)=\"1-sin%28x%29%2Fcos%28x%29\" and cos(x)/1+sin(x)=\"cos%28x%29%2F1%2Bsin%28x%29\" .
\n" ); document.write( "The parentheses are implicit in \"%281-sin%28x%29%29%2Fcos%28x%29=%281-sin%28x%29%29%2Fcos%28x%29%29\" .
\n" ); document.write( "The long horizontal line between numerator and denominator is a powerful grouping symbol that indicates that numerator and denominator must be calculated before doing the division.
\n" ); document.write( "When you cannot write two lines (as when you are entering calculations into a calculator or spreadsheet) you need to write the implicit parentheses.
\n" ); document.write( "
\n" ); document.write( "The expressions in \"%281-sin%28x%29%29%2Fcos%28x%29=cos%28x%29%2F%281%2Bsin%28x%29%29\" do not exist (are undefined) when the denominators are zero.
\n" ); document.write( "\"cos%28x%29=0\" happens for \"x=pi%2F2\" , \"x=3pi%2F2\" , and all co-terminal angles.
\n" ); document.write( "In general, we could say that \"x=%282k%2B1%29pi%2F2\" for every integer \"k\"
\n" ); document.write( "makes \"cos%28x%29=0\" and cannot be a solution to the equation above.
\n" ); document.write( "
\n" ); document.write( "The values of \"x\" that make \"1%2Bsin%28x%29=0\"<--->\"sin%28x%29=-1\" are
\n" ); document.write( "\"3pi%2F2\" and all co-terminal angles.
\n" ); document.write( "Those values will be excluded if we exclude \"x=%282k%2B1%29pi%2F2\" for every integer \"k\"
\n" ); document.write( "to make \"cos%28x%29%3C%3E0\" .
\n" ); document.write( "As long as \"x%3C%3E%282k%2B1%29pi%2F2\" for every integer \"k\" ,
\n" ); document.write( "which makes \"cos%28x%29%3C%3E0\" and \"cos%28x%29%3C%3E0\" , we can multiply both sides of
\n" ); document.write( "\"%281-sin%28x%29%29%2Fcos%28x%29=cos%28x%29%2F%281%2Bsin%28x%29%29\" times \"cos%28x%29%281%2Bsin%28x%29%29\"
\n" ); document.write( "to get the equivalent equations
\n" ); document.write( "--->\"%281-sin%28x%29%29%281%2Bsin%28x%29%29=cos%28x%29cos%28x%29\"--->\"1-sin%5E2%28x%29=cos%5E2%28x%29\" ,
\n" ); document.write( "and since the last equation is a trigonometric identity,
\n" ); document.write( "true for all values of \"x\" ,
\n" ); document.write( "the solution to \"1-sin%5E2%28x%29=cos%5E2%28x%29\" is all values of \"x\" ,
\n" ); document.write( "and the solution to \"%281-sin%28x%29%29%2Fcos%28x%29=cos%28x%29%2F%281%2Bsin%28x%29%29\"
\n" ); document.write( "is all values of \"x\" such that \"x%3C%3E%282k%2B1%29pi%2F2\" for every integer \"k\" . \n" ); document.write( "
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