In trigonometry, cofunctions (sine and cosine, cosecant and secant, and tangent and cotangent) of __________________________ ang
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What are 
and 
supposed to be? Can't answer a/b without that information.
I'll come back to part (c) in a moment. If we can show
is conservative, this part will be a breeze.
For part (d), to show whether is conservative, we have to show that there is a scalar function 
such that 
. It appears that you've written
I'm not sure what to make of the
that follows. means that
Integrating the first PDE with respect to
gives
Differentiating with respect to
gives
But we assumed that
is a function of alone, which means there is no solution for 
, and therefore no solution for . Hence is not conservative.
Back to part (c). is not conservative, so we have to compute the line integral the "long" way. Replacing 
and 
, we have
I'll come back to part (c) in a moment. If we can show
For part (d), to show whether
I'm not sure what to make of the
Integrating the first PDE with respect to
Differentiating with respect to
But we assumed that
Back to part (c).