Is the 1/2 supposed to be an exponent?

If so, you should have written it as

[sec(sinx)]^(1/2)

For written as [sec(sinx)]1/2, it is only multiplied by 1/2.

If it were only multiplied by 1/2, the domain would be



for we can take the sine of any angle x, and then take the secant of that.

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But if it's raised to the 1/2 power it is the same as a square root,
and we cannot take square roots of negative numbers in real number
mathematics.

Therefore we would have to omit from the domain all values where the 
secant is negative.

The secant is negative when the angle we're taking the secant of is in 
the 2nd or 3rd quadrants.

Therefore we would have to omit from the domain any values in the 2nd or
3rd quadrants, and include in the domain all values and only values which
are in the 1st or 4th quadrants.

We are taking secants of angles which are sines of other angles x. 

All sines are between -1 and +1, and values of angles between -1 and +1
are always in the 4th and 1st quadrant.  So the secants of these will always be 
positive.  Therefore the domain is all real numbers or



So whether the 1/2 was supposed to be an exponent or a multiplier, the
domain would be



either way.

Edwin