Step 1: Write the integral.
\[ I = \int \operatorname{cosec}(x-a)\operatorname{cosec} x\, dx \]
Step 2: Use a clever 1.
Multiply and divide by $\sin a$. The trick is to write $\sin a$ as $\sin[x - (x-a)]$.
\[ \sin a = \sin x \cos(x-a) - \cos x \sin(x-a) \]
Step 3: Split the fraction.
Dividing this expanded $\sin a$ by $\sin(x-a)\sin x$ gives two simple terms.
\[ \frac{\sin a}{\sin(x-a)\sin x} = \cot(x-a) - \cot x \]
Step 4: Rewrite the integral.
\[ I = \frac{1}{\sin a}\int [\cot(x-a) - \cot x]\, dx = \operatorname{cosec} a \int [\cot(x-a) - \cot x]\, dx \]
Step 5: Integrate each cot term.
\[ I = \operatorname{cosec} a\,[\log|\sin(x-a)| - \log|\sin x|] + c \]
Step 6: Combine the logs.
\[ I = \operatorname{cosec} a \cdot \log\left[\sin(x-a)\operatorname{cosec} x\right] + c \]
\[ \boxed{\operatorname{cosec} a \cdot \log[\sin(x-a)\operatorname{cosec} x] + c \text{ (Option 1)}} \]