document.write( "Question 1125669: By the principle of mathematical induction, prove that, for n ∈ N,
\n" ); document.write( "cos α + cos(α + β) + cos(α + 2β) + ЗЗЗ + cos(α + (n − 1)β) = cos (α+((n-1)β/2)).(sin(nβ/2))/(sin(nβ/2))
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Algebra.Com's Answer #741986 by MathLover1(20850) $\"\"$ $\"About$

You can put this solution on YOUR website!
Consider $\"P+%28n%29%3A+sin+%28alpha%29%2B+sin+%28alpha%2B+beta%29+%2B+sin+%28alpha+%2B+2beta%29\"$ + ... +

for all natural number $\"n\"$ .\r
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\n" ); document.write( "\n" ); document.write( "We observe that $\"P+%281%29\"$ is true, since
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\r
\n" ); document.write( "\n" ); document.write( "Assume that $\"P%28n%29\"$ is true for some natural numbers $\"k\"$ , i.e.,\r
\n" ); document.write( "\n" ); document.write( " $\"P+%28k%29+%3A+sin+%28alpha%29%2B+sin+%28alpha%2B+beta%29%2B+sin+%28alpha%2B+2beta%29\"$ + ... +

\r
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\n" ); document.write( "\n" ); document.write( "Now, to prove that $\"P%28k%2B+1%29\"$ is true, we have\r
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\n" ); document.write( "\n" ); document.write( " $\"P+%28k%2B+1%29+%3A+sin+%28alpha%29%2B+sin+%28alpha%2B+beta%29+%2B+sin+%28alpha%2B+2beta%29\"$ + ... +

\r
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\r
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\r
\n" ); document.write( "\n" ); document.write( "= $\"%28cos%28alpha+-beta%2F2%29-cos%28alpha+%2Bk%2Abeta%2Bbeta%2F2%29%29%2F2sin%28beta%2F2%29\"$ \r
\n" ); document.write( "\n" ); document.write( "= $\"%28sin%28alpha%2Bk%2Abeta%2F2%29%2Asin%28%28k%2Abeta%2Bbeta%29%2F2%29%29%2Fsin%28beta%2F2%29\"$ \r
\n" ); document.write( "\n" ); document.write( "= $\"%28sin%28alpha%2Bk%2Abeta%2F2%29%2Asin%28k%2B1%29%28beta%2F2%29%29%2Fsin%28beta%2F2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Thus $\"P+%28k%2B+1%29\"$ is $\"true\"$ whenever $\"P+%28k%29\"$ is $\"true\"$ .\r
\n" ); document.write( "\n" ); document.write( "Hence, by the Principle of Mathematical Induction $\"P%28n%29\"$ is true for all natural number $\"+n\"$ .\r
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