The smallest positive value of `x` (in radians) satisfying the equation `(log)_(cosx)((sqrt(3))/2sinx)=2-(log)_(secx)(tanx)` is (a)`pi/(12)` (b) `pi/6` (c) `pi/4` (d) `pi/3`

Explanation

`log_(cosx)(sqrt3/2sinx)=-log_(secx)(tanx)`

`log_(secx)(tanx)=(lntanx)/(lnsecx)=(lntanx)/(-lncosx)=-log_(cosx) tanx`

`log_cosxx(sqrt3/2sinx)=2+log_cosx(tanx)=log_cosx tanx`

`log_cosx(sqrt3/2cosx)=2`

`sqrt3/2cosx=cos^2x`

`cosx[cosx-sqrt3/2]=0`

`cosx=sqrt3/2`

`x=pi/6`.