document.write( "Question 822223: Prove . (sin x + cos x)(tan x + cot x) = sec x + csc x \n" ); document.write( "

Algebra.Com's Answer #494759 by math-vortex(648) $\"\"$ $\"About$

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document.write( "Hi, there--\r\n" );
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document.write( "YOUR PROBLEM:\r\n" );
document.write( "Prove that (sin x + cos x)(tan x + cot x) = sec x + csc x\r\n" );
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document.write( "A SOLUTION:\r\n" );
document.write( "In order to prove a trigonometric identity, we work on one side of the equation, rewriting it \r\n" );
document.write( "as a series of equivalent expressions until both sides of the equation are identical.\r\n" );
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document.write( "We have\r\n" );
document.write( "(sin x + cos x)(tan x + cot x) = sec x + csc x\r\n" );
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document.write( "Let's work on the left hand side. Apply the distributive property.\r\n" );
document.write( "(sin x)(tan x) + (sin x)(cot x) + (cos x)(tan x) + (cos x)(cot x) = sec x + csc x\r\n" );
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document.write( "Write tan x and cot x in terms of sin x and cos x\r\n" );
document.write( "(sin x)(sin x)/(cos x) + (sin x)(cos x)/(sin x) + (cos x)(sin x) /(cos x) + (cos x)(cos x) /(sin x) \r\n" );
document.write( "= sec x + csc x\r\n" );
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document.write( "Simplify, canceling terms where possible.\r\n" );
document.write( "(sin x)^2/(cos x) + (cos x) + (sin x) + (cos x)^2/(sin x) = sec x + csc x\r\n" );
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document.write( "Rewrite each term with a denominator of (sin x)( cos x).\r\n" );
document.write( "[(sin x)^2(sin x)]/[(sin x)(cos x)] + [(cos x)^2(sin x)]/[(sin x)(cos x)] + [(sin x)^2(cos x)]/[(sin x)\r\n" );
document.write( "(cos x)] + [(cos x)^2(cos x)]/[(sin x)(cos x)] = sec x + csc x\r\n" );
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document.write( "Factor sin x from the 1st and 2nd terms and cos x from the 3rd and 4th terms.\r\n" );
document.write( "(sin x)[(sin x)^2 + (cos x)^2)]/[(sin x)(cos x)] + (sin x)[(sin x)^2 + (cos x)^2)]/[(sin x)(cos x)] \r\n" );
document.write( "= sec x + csc x\r\n" );
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document.write( "Use the trig identify, (sin x)^2 + (cos x)^2 = 1, to simplify. Cancel like terms in numerator \r\n" );
document.write( "and denominator.\r\n" );
document.write( "([sin x + cos x]/[(sin x)(cos x)] = sec x + csc x\r\n" );
document.write( "(sin x)/[(sin x)(cos x)] + (cos x)/[(sin x)(cos x)] = sec x + csc x\r\n" );
document.write( "1/(cos x) + 1/(sin x) = sec x + csc x\r\n" );
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document.write( "Use the definition of sec x and csc x to rewrite the equation.\r\n" );
document.write( "sec x + csc x = sec x + sec x\r\n" );
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document.write( "That's it.\r\n" );
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document.write( "Hope that helps,\r\n" );
document.write( "Mrs. Figgy\r\n" );
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