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Question Number 117257 by bemath last updated on 10/Oct/20
Ifsin2(x)+cos2(x)=1thenwhatthevalueofsin11(x)+cos11(x)=?
Commented by bemath last updated on 10/Oct/20
nicequestion
Answered by mathmax by abdo last updated on 10/Oct/20
generallyletfindsin2n+1x+cos2n+1x=un(x)un(x)=(eix−e−ix2i)2n+1+(eix+e−ix2)2n+1=1(2i)2n+1∑k=02n+1C2n+1k(eix)k(−e−ix)2n+1−k+122n+1∑k=02n+1C2n+1k(eix)k(e−ix)2n+1−k=1(2i)2n+1∑k=02n+1C2n+1keikx(−1)2n+1−ke−i(2n+1−k)x+122n+1∑k=02n+1C2n+1keikxe−i(2n+1−k)x=−(−1)n22n+1i∑k=02n+1C2n+1kei(k−(2n+1)+k)x+122n+1∑k=02n+1C2n+1kei(k+k−(2n+1))x=i(−1)n22n+1∑k=02n+1C2n+1kei(2k−(2n+1))x+122n+1∑k=02n+1C2n+1kei(2k−(2n+1))x=(1+i(−1)n)×122n+1∑k=02n+1C2n+1kei(2k−(2n+1))xn=5⇒sin11x+cos11x=1−i211∑k=011C11kei(2k−11)x=1−i211{C110e−11ix+C111ei(−9)x+C112ei(−7)x+....C1111ei11x}
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