if matrix cos2pi/7 -sin2pi/7

sin2pi/7 cos 2pi/7 the whole power k = matrix 1 0

0 1 , then write the value of x+y+xy

cos2π7-sin2π7sin2π7cos2π7k=1001

Now, cos2π7-sin2π7sin2π7cos2π72=cos2π7-sin2π7sin2π7cos2π7×cos2π7-sin2π7sin2π7cos2π7

cos2π7-sin2π7sin2π7cos2π72=cos22π7-sin22π7   -2sin2π7cos2π72sin2π7cos2π7      cos22π7-sin22π7

cos2π7-sin2π7sin2π7cos2π72=cos2×2π7   -sin2×2π7sin2×2π7      cos2×2π7

Again, cos2π7-sin2π7sin2π7cos2π73=cos2π7-sin2π7sin2π7cos2π7×cos2π7-sin2π7sin2π7cos2π72

cos2π7-sin2π7sin2π7cos2π73=cos2π7-sin2π7sin2π7cos2π7×cos2×2π7   -sin2×2π7sin2×2π7      cos2×2π7

cos2π7-sin2π7sin2π7cos2π73=cos2π7cos2×2π7-sin2π7sin2×2π7             -sin2π7cos2×2π7-sin2×2π7cos2π7sin2π7cos2×2π7+sin2×2π7cos2π7                cos2π7cos2×2π7-sin2π7sin2×2π7

cos2π7-sin2π7sin2π7cos2π73=cos3×2π7-sin3×2π7sin3×2π7cos3×2π7

Similarly, if we go on, we will find that 

cos2π7-sin2π7sin2π7cos2π7k=cosk×2π7-sink×2π7sink×2π7cosk×2π7

i.e., cos2π7-sin2π7sin2π7cos2π7k=cos2kπ7-sin2kπ7sin2kπ7cos2kπ7  -------------------(1)

But, it is given that 

cos2π7-sin2π7sin2π7cos2π7k=1001 ------------------(2)

From (1) and (2), we have:-

cos2kπ7-sin2kπ7sin2kπ7cos2kπ7=1001

We are required to find the least positive integral value of k for which this holds true.

So, cos2kπ7=1, and, sin2kπ7=0

2kπ=0k=0

So, 0 is the least positive integral value of k.

  • -15

sry sry the ques is wrong... instead of "x+y+xy", "least positive inteagral value of k" has to come..

  • 3
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