Let `x= in (0,pi/2)dot` Then find the domain of the function `f(x)=1/(-(log)_(sinx)tanx)`

Explanation

Here, `x in (0,(pi)/(2))`
or `0 lt sinx lt 1 " " ` (1)
and we know `{(log_(a) x lt b, implies ,x gt a^(b)", if "0lt a lt 1),(,,x lt a^(b)", if " a gt 1):}} " " `(2)
Thus, `f(x)=(1)/(sqrt(-log_(sinx)tan x)) " exists if " -log_(sinx) (tanx) gt 0`
or `log_(sinx)tanx lt 0 " " `[ As inequality changes sign on multiplying by -ve]
or ` tanx gt (sinx)^(0) " " `[ Using (1) and (2) ]
or `tanx gt 1`
or `x in ((pi)/(4),(pi)/(2)) " "["As " x in (0,(pi)/(2))]`