document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1163196>Question 1163196</A>: <I><SPAN STYLE=\"word-break: break-all;\"> prove that the given equation is an identity\r<BR>\n" );
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document.write( "cos((pi/2)-x)=sinx </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #787231 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=787231\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> this can be solved geometrically, but you can also solve it using trig identities.\r<BR>\n" );
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document.write( "the trig identity you can use is:\r<BR>\n" );
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document.write( "cos(a - b) = cos(a) * cos(b) + sin(a) * sin(b)\r<BR>\n" );
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document.write( "the equation you want to prove is true is:\r<BR>\n" );
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document.write( "cos(pi/2) - x) = sin(x)\r<BR>\n" );
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document.write( "let a = pi/2 and b = x\r<BR>\n" );
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document.write( "the identity of:\r<BR>\n" );
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document.write( "cos(a - b) = cos(a) * cos(b) + sin(a) * sin(b) becomes:\r<BR>\n" );
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document.write( "cos(pi/2 - x) = cos(pi/2) * cos(x) + sin(pi/2) * sin(x)\r<BR>\n" );
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document.write( "cos(pi/2) = 0<BR>\n" );
document.write( "sin(pi/2) = 1\r<BR>\n" );
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document.write( "cos(pi/2 - x) = cos(pi/2) * cos(x) + sin(pi/2) * sin(x) becomes:<BR>\n" );
document.write( "cos(pi/2 - x) = 0 * cos(x) + 1 * sin(x)<BR>\n" );
document.write( "simplify  to get:<BR>\n" );
document.write( "cos(pi/2 - x) = sin(x)\r<BR>\n" );
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document.write( "there's your proof.\r<BR>\n" );
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document.write( "a nice list of trigonometric identities can be found at:<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.purplemath.com/modules/idents.htm\" target = \"_blank\">https://www.purplemath.com/modules/idents.htm</a>\r<BR>\n" );
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document.write( "the identity you are looking for will be under the title of angle sum and difference identities.\r<BR>\n" );
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document.write( "you can also solve this geometrically as follows:\r<BR>\n" );
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document.write( "draw a right triangle ABC with the right angle at C.<BR>\n" );
document.write( "since the sum of the angles of a triangle = 180, then:<BR>\n" );
document.write( "angle A + angle B + angle C = 180<BR>\n" );
document.write( "since angle C = 90 degrees, then:<BR>\n" );
document.write( "angle A + angle B + 90 degrees = 180 degrees<BR>\n" );
document.write( "subtract 90 from both sides of that equation to get:<BR>\n" );
document.write( "angle A + angle B = 90 degrees.<BR>\n" );
document.write( "solve for angle B to get:<BR>\n" );
document.write( "angle B = 90 - angle A.<BR>\n" );
document.write( "in the triangle ABC, the side opposite angle A is side a, the side opposite angle B is side b, the side opposite angle C is side c.<BR>\n" );
document.write( "since C is the 90 degree angle, then the hypotenuse of the triangle is side c.<BR>\n" );
document.write( "cos(angle B) = adjacent / hypotenuse = side a divided by side c<BR>\n" );
document.write( "sin(angle A) = opposite / hypotenuse = side a divided by side c<BR>\n" );
document.write( "this makes cos(angle B) = sin(angle A)<BR>\n" );
document.write( "since angle B = 90 - angle A, this makes cos(90 - angle A) = sin(angle A)<BR>\n" );
document.write( "let angle A = x, then this equation becomes:<BR>\n" );
document.write( "cos(90 - x) = sin(x)<BR>\n" );
document.write( "translate 90 degrees to radians to get:<BR>\n" );
document.write( "90 degrees * pi / 180 = pi/2 radians.<BR>\n" );
document.write( "replace 90 degrees with pi/2 radians to get cos(pi/2 - x) = sin(x)<BR>\n" );
document.write( "this also proves the equation is an identity.\r<BR>\n" );
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document.write( "here's a diagram that helps you to see what's happening with the geometric proof.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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