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Question

If 2cosx+sinx=1, then the sum of the values of 7cosx+6sinx is

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Solution

Using cosx=1tan2x21+tan2x2,sinx=2tanx21+tan2x2
sinx+2cosx=1
(2tanx21+tan2x2)+2(1tan2x21+tan2x2)=1
3tan2x22tanx21=0
(3tanx2+1)(tanx21)=0
tanx2=1 or 13
Now 7cosx+6sinx
=7(1tan2x21+tan2x2)+6(2tanx21+tan2x2)
When tanx2=17cosx+6sinx=6
And when tanx2=137cosx+6sinx=2
Therefore sum of value of solution is 6+2=8

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