- The given expression is a limit expression, where x tends to pi/2.
- We need to simplify the expression and find its value.



- We can use L'Hopital's rule to simplify the expression.
- Applying L'Hopital's rule, we get:

limit x tends to pi/2 (cosx - cosx^sinx * (1 + ln(sin(x))) / cosx
= limit x tends to pi/2 (1 - cosx^sinx * ln(sin(x))) / sinx

- Again applying L'Hopital's rule, we get:

limit x tends to pi/2 (cosx^sinx * (ln(cosx) - ln(sin(x)))^2 + cosx^sinx * (1/sin(x))) / cosx
= limit x tends to pi/2 (ln(cosx) - ln(sin(x)))^2 + (cosx^sinx / sinx)



- We can now directly substitute pi/2 in the expression to evaluate the limit.
- On substituting pi/2, we get:

limit x tends to pi/2 (ln(cosx) - ln(sin(x)))^2 + (cosx^sinx / sinx)
= (ln(0) - ln(1))^2 + (0/1)
= undefined

- Hence, the limit does not exist.



- The given limit does not exist as it evaluates to undefined on substituting pi/2 in the expression.
- Therefore, we cannot find its value.