Explanation

Simplify p = 1/(cosec θ + cot θ) − 1/sin θ as sin θ/(1 + cos θ) − 1/sin θ. Combining over the common denominator gives [sin²θ − (1 + cos θ)]/[sin θ(1 + cos θ)] = −cos θ/sin θ = −cot θ. The target expression 1/sin θ − 1/(cosec θ − cot θ) = 1/sin θ − sin θ/(1 − cos θ) simplifies identically to −cos θ/sin θ. Hence it equals p, option C.