document.write( "Question 223721: cosecant theta + cotangent theta * cosecant theta - cotangent theta = 1 \n" ); document.write( "
Algebra.Com's Answer #167221 by jsmallt9(3758)\"\" \"About 
You can put this solution on YOUR website!
(Algebra'com's formula software doesn't \"do\" theta's for some reason. So I will use t instead.)
\n" ); document.write( "Since this problem doesn't work out very well as you've written it, I going to assume that there are supposed to be parentheses:
\n" ); document.write( "\"%28csc%28t%29+%2B+cot%28t%29%29%28csc%28t%29+-+cot%28t%29%29+=+1\"
\n" ); document.write( "If this is wrong, stop reading.

\n" ); document.write( "When you don't see a better way to solve a problem in Trig., it is often helpful to rewrite the expressions in terms of sin and cos. So we will be replacing csc(t) with 1/sin(t) and cot(t) with cos(t)/sin(t):
\n" ); document.write( "
\n" ); document.write( "To multiply this it helps if you recognize that it fits the pattern: \"%28a%2Bb%29%28a-b%29+=+a%5E2+-+b%5E2\" with \"a\" being 1/sin(t) and \"b\" being cos(t)/sin(t). Using this pattern (or multiplying out the long way with FOIL and combining like terms) we get:
\n" ); document.write( "\"%281%2Fsin%28t%29%29%5E2+-+%28cos%28t%29%2Fsin%28t%29%29%5E2+=+1\"
\n" ); document.write( "which simplifies to:
\n" ); document.write( "\"1%2F%28sin%28t%29%29%5E2+-+%28cos%28t%29%29%5E2%2F%28sin%28t%29%29%5E2+=+1\"
\n" ); document.write( "The fractions have the same denominator so we can subtract them:
\n" ); document.write( "\"%281+-+%28cos%28t%29%29%5E2%29%2F%28sin%28t%29%29%5E2+=+1\"
\n" ); document.write( "Since \"1+-+%28cos%28t%29%29%5E2+=+%28sin%28t%29%29%5E2\" the numerator becomes:
\n" ); document.write( "\"%28sin%28t%29%29%5E2%2F%28sin%28t%29%29%5E2+=+1\"
\n" ); document.write( "And the fraction cancels leaving
\n" ); document.write( "\"1+=+1\"
\n" ); document.write( "And with this final equation, since 1 always equal 1, shows that your original equation is also always true. We call this an identity. \n" ); document.write( "
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