document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=602392>Question 602392</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi. Please help me prove that this equation is an identity because I really don't know what to do. Thank you!\r<BR>\n" );
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document.write( "(cot B / sec B - tan B)  -  (cos B / sec B + tan B)  =  sin B + csc B </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #380216 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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/cgi-bin/show-face.mpl?user=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=380216\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{\cot B}{\sec B\ -\ \tan B}\ -\ \frac{\cos B}{\sec B\ +\ \tan B}\ \equiv^?\ \sin B\ +\ \csc B\">\r<BR>\n" );
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document.write( "The LCD in the LHS is the product of the two denominators.  The two denominators are a conjugate pair, hence the product is the difference of two squares.  Apply the LCD:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{(\sec B\ +\ \tan B)\cot B\ -\ (\sec B\ -\ \tan B)\cos B}{\sec^2 B\ -\ \tan^2 B}\ \equiv^?\ \sin B\ +\ \csc B\">\r<BR>\n" );
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document.write( "Use the fact that tangent is sine over cosine, cotangent is the reciprocal of tangent, secant is the reciprocal of cosine, and cosecant is the reciprocal of sine to write:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{(\frac{1}{\cos B}\ +\ \frac{\sin B}{\cos B})\frac{\cos B}{\sin B}\ -\ (\frac{1}{\cos B}\ -\ \frac{\sin B}{\cos B})\cos B}{\frac{1}{\cos^2 B}\ -\ \frac{\sin^2 B}{\cos^2 B}}\ \equiv^?\ \sin B\ +\ \csc B\">\r<BR>\n" );
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document.write( "Simplify the LHS numerator:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{(\frac{1}{\sin B}\ +\ 1)\ -\ (1\ -\ \sin B)}{\frac{1}{\cos^2 B}\ -\ \frac{\sin^2 B}{\cos^2 B}}\ \equiv^?\ \sin B\ +\ \csc B\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{\frac{1}{\sin B}\ +\ \sin B}{\frac{1}{\cos^2 B}\ -\ \frac{\sin^2 B}{\cos^2 B}}\ \equiv^?\ \sin B\ +\ \csc B\">\r<BR>\n" );
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document.write( "Combine the fractions in the LHS denominator:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{\frac{1}{\sin B}\ +\ \sin B}{\frac{1\ -\ \sin^2 B}{\cos^2 B}}\ \equiv^?\ \sin B\ +\ \csc B\">\r<BR>\n" );
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document.write( "Use the Pythagorean identity in the LHS denominator:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{1}{\sin B}\ +\ \sin B\ \equiv^?\ \sin B\ +\ \csc B\">\r<BR>\n" );
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document.write( "Finally, apply cosecant is the reciprocal of sine\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \csc B\ +\ \sin B\ \equiv \ \sin B\ +\ \csc B\">\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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