document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1198384>Question 1198384</A>: <I><SPAN STYLE=\"word-break: break-all;\"> After transformations, the expression sin (A + B)sin (A - B) can be proved to be equal to:<BR>\n" );
document.write( "A) sin^2 A - sin^2 <BR>\n" );
document.write( "B) sin A - sin B  <BR>\n" );
document.write( "C) 2 sin A sin B<BR>\n" );
document.write( "D) 2 cos A cos B<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #831963 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=831963\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answer: <font color=red>sin^2(A) - sin^2(B)</font>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "================================================================================\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Explanation:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "There are at least two methods we can follow to prove that <BR>\n" );
document.write( "sin(A+B)sin(A-B) = sin^2(A)-sin^2(B)<BR>\n" );
document.write( "is an identity\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "---------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Method 1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let<BR>\n" );
document.write( "m = sin(A)cos(B)<BR>\n" );
document.write( "n = cos(A)sin(B)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "sin(A+B)sin(A-B) = [ sin(A)cos(B)+cos(A)sin(B) ] * [ sin(A)cos(B)-cos(A)sin(B) ]<BR>\n" );
document.write( "sin(A+B)sin(A-B) = (m+n)(m-n)<BR>\n" );
document.write( "sin(A+B)sin(A-B) = m^2 - n^2<BR>\n" );
document.write( "sin(A+B)sin(A-B) = (sin(A)cos(B))^2 - (cos(A)sin(B))^2<BR>\n" );
document.write( "sin(A+B)sin(A-B) = sin^2(A)cos^2(B) - cos^2(A)sin^2(B)<BR>\n" );
document.write( "sin(A+B)sin(A-B) = sin^2(A)(1-sin^2(B)) - (1-sin^2(A))sin^2(B)<BR>\n" );
document.write( "sin(A+B)sin(A-B) = sin^2(A)-sin^2(A)sin^2(B) - sin^2(B)+sin^2(A)sin^2(B)<BR>\n" );
document.write( "sin(A+B)sin(A-B) = sin^2(A) - sin^2(B)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The relevant identities that I used were:<BR>\n" );
document.write( "sin(A+B) = sin(A)cos(B)+cos(A)sin(B)<BR>\n" );
document.write( "sin(A-B) = sin(A)cos(B)-cos(A)sin(B)<BR>\n" );
document.write( "cos^2(A) = 1 - sin^2(A) which is from sin^2(A)+cos^2(A) = 1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "---------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Method 2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Another identity to use is<BR>\n" );
document.write( "sin(u)*sin(v) = 0.5 * [ cos(u-v) - cos(u+v) ]<BR>\n" );
document.write( "which should be somewhere on your list of trig identities.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let<BR>\n" );
document.write( "u = A+B<BR>\n" );
document.write( "v = A-B<BR>\n" );
document.write( "So,<BR>\n" );
document.write( "sin(u)*sin(v) = 0.5 * [ cos(u-v) - cos(u+v) ]<BR>\n" );
document.write( "sin(A+B)*sin(A-B) = 0.5 * [ cos(A+B-(A-B)) - cos(A+B+A-B) ]<BR>\n" );
document.write( "sin(A+B)*sin(A-B) = 0.5 * [ cos(2B) - cos(2A) ]<BR>\n" );
document.write( "sin(A+B)*sin(A-B) = 0.5 * [ (1-2sin^2(B))-(1-2sin^2(A)) ]<BR>\n" );
document.write( "sin(A+B)*sin(A-B) = 0.5 * [ 1-2sin^2(B)-1+2sin^2(A) ]<BR>\n" );
document.write( "sin(A+B)*sin(A-B) = 0.5 * [ -2sin^2(B)+2sin^2(A) ]<BR>\n" );
document.write( "sin(A+B)*sin(A-B) = 0.5 * [ 2sin^2(A)-2sin^2(B) ]<BR>\n" );
document.write( "sin(A+B)*sin(A-B) = sin^2(A)-sin^2(B)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Edit:<BR>\n" );
document.write( "Here's a handy list of trig identities<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://tutorial.math.lamar.edu/pdf/Trig_Cheat_Sheet.pdf\">https://tutorial.math.lamar.edu/pdf/Trig_Cheat_Sheet.pdf</a><BR>\n" );
document.write( "</font><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );