The value of tan 1° tan 2º tan 3° .... tan 89° is :

Q: The value of tan 1° tan 2º tan 3° .... tan 89° is :

The problem requires us to find the value of the product $ \tan 1^\circ \tan 2^\circ \tan 3^\circ \ldots \tan 89^\circ $. ### Explanation for Option 1 (Correct Answer: 1) The key to solving this problem lies in understanding the properties of the tangent function and the symmetry of angles in a right triangle. The tangent function has a periodicity of 180°, and it is known that: \[ \tan(90^\circ - \theta) = \cot(\theta) \] This means that for any angle $\theta$, $\tan(90^\circ - \theta)$ is the reciprocal of $\tan(\theta)$. Therefore, in the sequence of angles from 1° to 89°, each angle $\theta$ has a complementary angle $90^\circ - \theta$ such that: \[ \tan(\theta) \cdot \tan(90^\circ - \theta) = 1 \] For example: - $\tan 1^\circ \cdot \tan 89^\circ = 1$ - $\tan 2^\circ \cdot \tan 88^\circ = 1$ - $\tan 3^\circ \cdot \tan 87^\circ = 1$ This pattern continues for all pairs of angles from 1° to 89°. Since each pair of angles multiplies to 1, the entire product: \[ \tan 1^\circ \tan 2^\circ \tan 3^\circ \ldots \tan 89^\circ = 1 \] Thus, the correct answer is Option 1: 1. ### Explanation for Why Other Options are Incorrect - **Option 2 (-1):** The product of the tangents of complementary angles is positive, as shown above. Since each pair multiplies to 1, the overall product cannot be negative. - **Option 3 (0):** The tangent function is never zero for angles between 0° and 90° (exclusive). Therefore, none of the terms in the product is zero, and the product itself cannot be zero. - **Option 4 ($\frac{1}{\sqrt{2}}$):** This value does not arise from the product of the tangents of complementary angles. As shown, each pair of angles multiplies to 1, and there is no mathematical basis for the product to equal $\frac{1}{\sqrt{2}}$. In conclusion, the correct answer is Option 1: 1, due to the complementary angle property of the tangent function.

Question

Accepted Answer

The problem requires us to find the value of the product $ \tan 1^\circ \tan 2^\circ \tan 3^\circ \ldots \tan 89^\circ $.

### Explanation for Option 1 (Correct Answer: 1)

The key to solving this problem lies in understanding the properties of the tangent function and the symmetry of angles in a right triangle. The tangent function has a periodicity of 180°, and it is known that:

$$
\tan(90^\circ - \theta) = \cot(\theta)
$$

This means that for any angle $\theta$, $\tan(90^\circ - \theta)$ is the reciprocal of $\tan(\theta)$. Therefore, in the sequence of angles from 1° to 89°, each angle $\theta$ has a complementary angle $90^\circ - \theta$ such that:

$$
\tan(\theta) \cdot \tan(90^\circ - \theta) = 1
$$

For example:
- $\tan 1^\circ \cdot \tan 89^\circ = 1$
- $\tan 2^\circ \cdot \tan 88^\circ = 1$
- $\tan 3^\circ \cdot \tan 87^\circ = 1$

This pattern continues for all pairs of angles from 1° to 89°. Since each pair of angles multiplies to 1, the entire product:

$$
\tan 1^\circ \tan 2^\circ \tan 3^\circ \ldots \tan 89^\circ = 1
$$

Thus, the correct answer is Option 1: 1.

### Explanation for Why Other Options are Incorrect

- **Option 2 (-1):** The product of the tangents of complementary angles is positive, as shown above. Since each pair multiplies to 1, the overall product cannot be negative.

- **Option 3 (0):** The tangent function is never zero for angles between 0° and 90° (exclusive). Therefore, none of the terms in the product is zero, and the product itself cannot be zero.

- **Option 4 ($\frac{1}{\sqrt{2}}$):** This value does not arise from the product of the tangents of complementary angles. As shown, each pair of angles multiplies to 1, and there is no mathematical basis for the product to equal $\frac{1}{\sqrt{2}}$.

In conclusion, the correct answer is Option 1: 1, due to the complementary angle property of the tangent function.