document.write( "Question 552130: Prove the following identity:\r
\n" ); document.write( "\n" ); document.write( "(1 + sin x + cos x)/(1 + sin x - cos x) = cot(x/2) \n" ); document.write( "
Algebra.Com's Answer #360147 by AnlytcPhil(1807)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "A familiar identity is  tan\"%28x%2F2%29\" = \"%281-cos%28x%29%29%2Fsin%28x%29\", and\r\n" );
document.write( "since the cotangent is the reciprocal of the tangent, we have\r\n" );
document.write( "cot\"%28x%2F2%29\" = \"sin%28x%29%2F%281-cos%28x%29%29\",  so we will try to make\r\n" );
document.write( "the left side into that expression. So we will multiply the left side\r\n" );
document.write( "by \"%281-cos%28x%29%29%2F%281-cos%28x%29%29\", so hopefully it will have the desired denominator in the end. \r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "1 + sin(x) + cos(x)\r\n" );
document.write( "——————————————————— = cot(\"x%2F2\")\r\n" );
document.write( "1 + sin(x) - cos(x)  \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "[1 + sin(x) + cos(x)]   [1 - cos(x)] \r\n" );
document.write( "————————————————————— · ————————————\r\n" );
document.write( "[1 + sin(x) - cos(x)]   [1 - cos(x)]\r\n" );
document.write( "\r\n" );
document.write( "1 - cos(x) + sin(x) - sin(x)cos(x) + cos(x) - cos²(x) \r\n" );
document.write( "—————————————————————————————————————————————————————\r\n" );
document.write( "       [1 + sin(x) - cos(x)][1 - cos(x)]\r\n" );
document.write( "\r\n" );
document.write( "1 - cos(x) + sin(x) - sin(x)cos(x) + cos(x) - cos²(x) \r\n" );
document.write( "—————————————————————————————————————————————————————\r\n" );
document.write( "       [1 + sin(x) - cos(x)][1 - cos(x)]\r\n" );
document.write( "\r\n" );
document.write( "      1 + sin(x) - sin(x)cos(x) - cos²(x) \r\n" );
document.write( "      ———————————————————————————————————\r\n" );
document.write( "       [1 + sin(x) - cos(x)][1 - cos(x)]\r\n" );
document.write( "\r\n" );
document.write( "Rearrange the terms\r\n" );
document.write( "\r\n" );
document.write( "      1 - cos²(x) + sin(x) - sin(x)cos(x)  \r\n" );
document.write( "     —————————————————————————————————————\r\n" );
document.write( "       [1 + sin(x) - cos(x)][1 - cos(x)]\r\n" );
document.write( "\r\n" );
document.write( "     [1 - cos²(x)] + sin(x) - sin(x)cos(x)  \r\n" );
document.write( "     —————————————————————————————————————\r\n" );
document.write( "       [1 + sin(x) - cos(x)][1 - cos(x)]\r\n" );
document.write( "  \r\n" );
document.write( "Use the identity sin²(x) = 1 - cos²(x)\r\n" );
document.write( "\r\n" );
document.write( "        sin²(x) + sin(x) - sin(x)cos(x)  \r\n" );
document.write( "       —————————————————————————————————\r\n" );
document.write( "       [1 + sin(x) - cos(x)][1 - cos(x)]\r\n" );
document.write( "\r\n" );
document.write( "Factor out sin(x) on the top:\r\n" );
document.write( "\r\n" );
document.write( "         sin(x)[sin(x) + 1 - cos(x)]  \r\n" );
document.write( "       —————————————————————————————————\r\n" );
document.write( "       [1 + sin(x) - cos(x)][1 - cos(x)]\r\n" );
document.write( "\r\n" );
document.write( "Then we can cancel:\r\n" );
document.write( "\r\n" );
document.write( "         sin(x)[sin(x) + 1 - cos(x)]  \r\n" );
document.write( "       —————————————————————————————————\r\n" );
document.write( "       [1 + sin(x) - cos(x)][1 - cos(x)]\r\n" );
document.write( " \r\n" );
document.write( "                  sin(x)  \r\n" );
document.write( "                ——————————\r\n" );
document.write( "                1 - cos(x)\r\n" );
document.write( "\r\n" );
document.write( "which is the identity we showed above for cot(\"x%2F2\")\r\n" );
document.write( "\r\n" );
document.write( "                cot\"%28x%2F2%29\"\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );