NIMCET 2008 — Mathematics PYQ

Step 1: Identify the Key Trigonometric Identity

We use the identity for $\tan (A+B)$. Specifically, if $A+B=45^{\circ }$, then:

Now, add $1$ to both sides:

Summary: If $A+B=45^{\circ }$, then $(1+\tan A)(1+\tan B)=2$.

Step 2: Pair the terms in the series

The given expression is:

We can pair the terms where the angles sum to $45^{\circ }$:

• $(1+\tan 1^{\circ })(1+\tan 44^{\circ })=2$

• $(1+\tan 2^{\circ })(1+\tan 43^{\circ })=2$

• ... and so on.

Step 3: Count the pairs

The terms from $1^{\circ }$ to $44^{\circ }$ consist of $44$ individual terms. Since we are pairing them, there are:

Each pair results in a value of $2$. So, the product of these $44$ terms is $2^{22}$.

Step 4: Consider the final term

The last term in the series is $(1+\tan 45^{\circ })$.

Since $\tan 45^{\circ }=1$:

Step 5: Calculate the total product

Total Product $=(\text{Product of 22 pairs})\times (1+\tan 45^{\circ })$

Total Product $=2^{22}\times 2^1=2^{23}$

Comparing this with the given $2^n$:

Conclusion:

The value of $n$ is 23.

Correct Option: (c)