The number of values of x in [0,npi] ,n...

The number of values of x `in [0,npi] ,n in Z` that satisfy the equation ` log_|sinx|(1+cosx)=2` is

none of these

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The correct Answer is:

We observe that `"log"_(|"sin"x|) (1+"cos"x) " is defined if " x ne n pi, (2n +1) (pi)/(2), n in Z`.
Now,
`"log"_(|"sin"x|) (1+"cos"x)=2`
`rArr 1+"cos" x= |"sin"x|^(2)`
`rArr "cos"^(2)x + "cos" x = 0 rArr "cos" x (1+"cos" x) = 0`
But, `"cos"^(2)x + "cos" x ne 0 " for any " x in (0, n pi) - (2n-1) (pi)/(2), n in Z`.
Hence, the given equation has no solution.

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