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Question

If F(x)=cosxsinx0sinxcosx0001, then show that F (x)F(y)=F(x+y).

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Solution

LHS=F(x)F(y)=cosxsinx0sinxcosx0001cosysiny0sinycosy0001=cosxcosysinxsinysinycosxsinxcosy0sinxcosy+cosxsinysinxsiny+cosxcosy0001

[cos(A+B)=cosAcosBsinAsinBsin(A+B)=sinAcosB+sinBcosA]=cos(x+y)sin(x+y)0sin(x+y)cos(x+y)0001
Now, replacing x by (x+y)in F(x) F(y)
F(x+y)=cos(x+y)sin(x+y)0sin(x+y)cos(x+y)0001F(x)F(y)=F(x+y)=RHS. Hence, proved.


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