amath..urgent

用戶18025

2009-01-24 5:42 am

Prove that sinxcosnx=sin nxcosx-sin(n-1)x for all intergers n .
Hence prove that d/dxcos^mxcosnx=mcos^(m-1)xsin(n-1)x-(m+n)cos^mxsin nx.for any integer m greater or equal to 2

回答 (1)

myisland8132

2009-01-24 6:09 am

✔ 最佳答案

Part 1
ehwn n=1
LHS=sinxcosx
RHS=sinxcosx
LHS=RHS when n=1, the statement is true
Assume that when n=k, the statement is true
i.e. sinxcoskx=sin kxcosx-sin(k-1)x
when n=k+1
LHS
=sinxcos(k+1)x
=sinx[coskxcosx-sinkxsinx]
=sinxcosxcoskx-sin^2xsinkx
=sinxcosxcoskx-(1-cos^2x)sinkx
=sinxcosxcoskx+cos^2xsinkx-sinkx
=cosx(sinxcoskx+cosxsinkx)-sinkx
=sin(k+1)xcosx-sinkx
=RHS
So when n=k+1,the statement is true
By MI, for all intergers n, sinxcosnx=sin nxcosx-sin(n-1)x
d/dx(cos^mxcosnx)
=-ncos^mxsinnx-mcosnxcos^(m-1)xsinx
=-ncos^mxsinnx-mcos^(m-1)x[sin nxcosx-sin(n-1)x]
=mcos^(m-1)xsin(n-1)x-(m+n)cos^mxsin nx.for any integer m greater or equal to 2

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