NIMCET 2022 — Mathematics PYQ

Solution

Given $a_1,a_2,\dots ,a_n$ are in A.P., we have:

$d=a_2-a_1=a_3-a_2=\cdots =a_n-a_{n-1}$

The given expression is:

Step 1: Distribute $\sin d$ into each term

Step 2: Replace $d$ in each term with the corresponding difference

Step 3: Apply the identity $\sin (A-B)=\sin A\cos B-\cos A\sin B$

For the first term:

Step 4: Expand the entire series

Step 5: Cancel the intermediate terms (Telescoping Sum)

All terms cancel out except the first and the last:

Final Answer:

The sum is equal to $\cot a_1-\cot a_n$. (Option B)