document.write( "Question 800722: Prove that tan(10°)cos(20°)tan(40°)tan(50°)csc(70°)sec(80°)sin(80°) = 1 \n" ); document.write( "
Algebra.Com's Answer #483243 by Edwin McCravy(20056)\"\" \"About 
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document.write( "Prove that tan(10°)cos(20°)tan(40°)tan(50°)csc(70°)sec(80°)sin(80°) = 1\r\n" );
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document.write( "Change all trig ratios other than the sines and cosines\r\n" );
document.write( "to sines and cosines, and put 1 denominators under the \r\n" );
document.write( "sines and cosines, using these quotient and reciprocal \r\n" );
document.write( "identities: \r\n" );
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document.write( "\"tan%28theta%29%29\"\"%22%22=%22%22\"\"sin%28theta%29%2Fcos%28theta%29\", \"sec%28theta%29\"\"%22%22=%22%22\"\"1%2Fcos%28theta%29\", \"csc%28theta%29\"\"%22%22=%22%22\"\"1%2Fsin%28theta%29\"\r\n" );
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document.write( "\"sin%28%2210%B0%22%29%2Fcos%28%2210%B0%22%29\"\"%22%22%2A%22%22\"\"cos%28%2220%B0%22%29%2F1\"\"%22%22%2A%22%22\"\"sin%28%2240%B0%22%29%2Fcos%28%2240%B0%22%29\"\"%22%22%2A%22%22\"\"sin%28%2250%B0%22%29%2Fcos%28%2250%B0%22%29\"\"%22%22%2A%22%22\"\"1%2Fsin%28%2270%B0%22%29\"\"%22%22%2A%22%22\"\"1%2Fcos%28%2280%B0%22%29\"\"%22%22%2A%22%22\"\"sin%28%2280%B0%22%29%2F1\"\"%22%22%2A%22%22\"\r\n" );
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document.write( "Next change all the cosines to sines using the cofuction identity:\r\n" );
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document.write( "\"cos%28theta%29\"\"%22%22=%22%22\"\"sin%28%2290%B0%22-theta%29\"\r\n" );
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document.write( "\"cos%28%2210%B0%22%29\"\"%22%22=%22%22\"\"sin%28%2290%B0%22-%2210%B0%22%29\"\"%22%22=%22%22\"\"sin%28%2280%B0%22%29\"\r\n" );
document.write( "\"cos%28%2220%B0%22%29\"\"%22%22=%22%22\"\"sin%28%2290%B0%22-%2220%B0%22%29\"\"%22%22=%22%22\"\"sin%28%2270%B0%22%29\"\r\n" );
document.write( "\"cos%28%2240%B0%22%29\"\"%22%22=%22%22\"\"sin%28%2290%B0%22-%2240%B0%22%29\"\"%22%22=%22%22\"\"sin%28%2250%B0%22%29\"\r\n" );
document.write( "\"cos%28%2250%B0%22%29\"\"%22%22=%22%22\"\"sin%28%2290%B0%22-%2250%B0%22%29\"\"%22%22=%22%22\"\"sin%28%2240%B0%22%29\"\r\n" );
document.write( "\"cos%28%2280%B0%22%29\"\"%22%22=%22%22\"\"sin%28%2290%B0%22-%2280%B0%22%29\"\"%22%22=%22%22\"\"sin%28%2210%B0%22%29\"\r\n" );
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document.write( "Substitute those for all the cosines:\r\n" );
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document.write( "\"sin%28%2210%B0%22%29%2Fsin%28%2280%B0%22%29\"\"%22%22%2A%22%22\"\"sin%28%2270%B0%22%29%2F1\"\"%22%22%2A%22%22\"\"sin%28%2240%B0%22%29%2Fsin%28%2250%B0%22%29\"\"%22%22%2A%22%22\"\"sin%28%2250%B0%22%29%2Fsin%28%2240%B0%22%29\"\"%22%22%2A%22%22\"\"1%2Fsin%28%2270%B0%22%29\"\"%22%22%2A%22%22\"\"1%2Fsin%28%2210%B0%22%29\"\"%22%22%2A%22%22\"\"sin%28%2280%B0%22%29%2F1\"\"%22%22%2A%22%22\"\r\n" );
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document.write( "That all cancels and gives 1\r\n" );
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document.write( "Edwin
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