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Question

If (2cosx+sinx)=1, then sum of all possible value of (7cosx+6sinx) is__________

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Solution

Given 2cosx+sinx=12(1tan2x2)1+tan2x2+2tanx21+tan2x2=1

22tan2x2+2tanx2=1+tan2x2

3tan2x22tanx21=0

3tan2x23tanx2+tanx21=0

(tanx21)(3tanx2+1)=0

So tanx2=1,13

When tanx2=1
then 7cosx+6sinx=7(1tan2x21+tan2x2)+6(2tanx21+tan2x2)=7×111+1+6×2×11+1=0+6=6

When tanx2=13
then 7cosx+6sinx=7(1tan2x21+tan2x2)+6(2tanx21+tan2x2)=7×1191+19+6×2×131+19=7×8106×610=2
The sum of all possible values is 6+2=8

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