Solve `(2^(x)-1)(3^(x)-9)(sinx-cosx)(5^(x)-1)lt0, -pi//2 lt x lt 2pi`.

Explanation

We have
`(2^(x)-1)(3^(x)-9)(sinx-cosx)(5^(x)-1)lt0, -pi//2 lt x lt 2pi`.
`2^(x)-1=0 " if " x=0`
`3^(x)-9=0" if "x=2`
`sinx-cosx=0" if " tanx=1`
` "or "x=(pi)/(4),(5pi)/(4)`
`5^(x)-1=0" if " x=0`
The sign scheme of expression `(2^(x)-1)(3^(x)-9)(sinx-cosx)(5^(x)-1)` is as shown in the following figure.

When `x in (-pi//2,0),` each of the factors in the product is negative.
So, expression is positive for `x in (-pi//2,0)`.
At x = 0, sign is not changing as two factors vanishes at `x = 0, 2^(x)-1` and `5^(x)-1`.
From the sign scheme, we have `(2^(x)-1)(3^(x)-9)(sinx-cosx)(5^(x)-1) lt 0` for
`x in (pi//4,2)cup (5pi//4,2pi)`