document.write( "Question 594379: Prove that \"1+sin^2x=csc^2x-cot^2x+sin^2x\" \n" ); document.write( "

Algebra.Com's Answer #376764 by jsmallt9(3758) $\"\"$ $\"About$

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$\"1%2Bsin%5E2%28x%29=csc%5E2%28x%29-cot%5E2%28x%29%2Bsin%5E2%28x%29\"$
\n" ); document.write( "There are many techniques to proving identities (equations that are true for all x's. One technique is to match terms. We already have a match (which I have highlighted in red):
\n" ); document.write( " $\"1%2Bred%28sin%5E2%28x%29%29=csc%5E2%28x%29-cot%5E2%28x%29%2Bred%28sin%5E2%28x%29%29\"$
\n" ); document.write( "We will leave the matched terms alone. Now we must find a way to show that the remaining terms on each side are equal. The unmatched terms are 1, $\"csc%5E2%28x%29\"$ and $\"cot%5E2%28x%29\"$ . If we know our properties well we know that one of the Pythagorean properties has exactly those three terms:
\n" ); document.write( " $\"1+%2B+cot%5E2%28x%29+=+csc%5E2%28x%29\"$ . If we subtract $\"cot%5E2%28x%29\"$ from each side we get:
\n" ); document.write( " $\"1+=+csc%5E2%28x%29+-+cot%5E2%28x%29\"$
\n" ); document.write( "So we can substitute the right side for the 1 in your equation giving us:
\n" ); document.write( "

\n" ); document.write( "And we are finished. \n" ); document.write( "

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