The value of $\frac{sin~55^{\circ}}{cos~35^{\circ}}+\frac{cot~1^{\circ}cot~2^{\circ}cot~3^{\circ}......cot~90^{\circ}}{1+2+3+......+90}$ is:

Question

Accepted Answer

To solve the given problem, we need to evaluate the expression:

$$
\frac{\sin 55^\circ}{\cos 35^\circ} + \frac{\cot 1^\circ \cot 2^\circ \cot 3^\circ \ldots \cot 90^\circ}{1 + 2 + 3 + \ldots + 90}
$$

Let's break this down into two parts:

### Part 1: $\frac{\sin 55^\circ}{\cos 35^\circ}$

Using the trigonometric identity $\sin(90^\circ - \theta) = \cos \theta$, we have:

$$
\sin 55^\circ = \cos(90^\circ - 55^\circ) = \cos 35^\circ
$$

Thus,

$$
\frac{\sin 55^\circ}{\cos 35^\circ} = \frac{\cos 35^\circ}{\cos 35^\circ} = 1
$$

### Part 2: $\frac{\cot 1^\circ \cot 2^\circ \ldots \cot 90^\circ}{1 + 2 + 3 + \ldots + 90}$

The product $\cot 1^\circ \cot 2^\circ \ldots \cot 90^\circ$ includes $\cot 90^\circ$, which is undefined because $\cot 90^\circ = \frac{\cos 90^\circ}{\sin 90^\circ} = \frac{0}{1} = 0$. Therefore, the entire product is zero.

The sum $1 + 2 + 3 + \ldots + 90$ is a simple arithmetic series, which can be calculated using the formula for the sum of the first $n$ natural numbers:

$$
S = \frac{n(n + 1)}{2}
$$

For $n = 90$:

$$
S = \frac{90 \times 91}{2} = 4095
$$

Thus, the second part of the expression becomes:

$$
\frac{0}{4095} = 0
$$

### Final Evaluation

Combining both parts, we have:

$$
1 + 0 = 1
$$

Therefore, the value of the entire expression is 1, which corresponds to Option 2.

### Explanation of Incorrect Options

- **Option 1 (-1):** The expression evaluates to 1, not -1. There is no subtraction or negative value involved in the calculation.
- **Option 3 (0):** While the second part of the expression evaluates to 0, the first part evaluates to 1, making the total 1, not 0.
- **Option 4 (not defined):** Although $\cot 90^\circ$ is undefined, it results in a product of zero, which is a defined value. The overall expression is well-defined and evaluates to 1.

Thus, Option 2 is indeed the correct answer.