If $ riangle$ ABC is a right angled at A, then value of tan B × tan C is :

Q: If $\triangle$ ABC is a right angled at A, then value of tan B × tan C is :

In a right-angled triangle, such as $\triangle ABC$ where $\angle A$ is the right angle, the trigonometric functions of the angles $\angle B$ and $\angle C$ have specific relationships. Since $\angle A$ is $90^\circ$, the sum of the other two angles, $\angle B$ and $\angle C$, must be $90^\circ$ as well. This means $\angle B + \angle C = 90^\circ$. The tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side. For angles $\angle B$ and $\angle C$, we have: \[ \tan B = \frac{\text{opposite side to } B}{\text{adjacent side to } B} \] \[ \tan C = \frac{\text{opposite side to } C}{\text{adjacent side to } C} \] Since $\angle B + \angle C = 90^\circ$, $\angle C$ is the complement of $\angle B$. In trigonometry, the tangent of complementary angles has a specific relationship: \[ \tan(90^\circ - \theta) = \cot \theta \] Thus, $\tan C = \cot B$. Therefore, the product of $\tan B$ and $\tan C$ is: \[ \tan B \times \tan C = \tan B \times \cot B = \tan B \times \frac{1}{\tan B} = 1 \] This confirms that the value of $\tan B \times \tan C$ is indeed 1, which supports Option 3 as the correct answer. Now, let's explain why the other options are incorrect: 1. **Option 1: -1** - The product of $\tan B$ and $\tan C$ cannot be negative because both angles are acute angles in a right triangle, and the tangent of acute angles is positive. 2. **Option 2: 0** - The product $\tan B \times \tan C$ cannot be zero unless one of the tangents is zero. However, in a right triangle, neither $\angle B$ nor $\angle C$ can be $0^\circ$ or $90^\circ$, so their tangents cannot be zero. 3. **Option 4: $\infty$** - The product $\tan B \times \tan C$ cannot be infinite because both $\tan B$ and $\tan C$ are finite positive values for acute angles in a right triangle. Therefore, the correct answer is indeed Option 3, as the product of $\tan B$ and $\tan C$ is 1.

Question

If $	riangle$ ABC is a right angled at A, then value of tan B × tan C is :

Accepted Answer

In a right-angled triangle, such as $\triangle ABC$ where $\angle A$ is the right angle, the trigonometric functions of the angles $\angle B$ and $\angle C$ have specific relationships. Since $\angle A$ is $90^\circ$, the sum of the other two angles, $\angle B$ and $\angle C$, must be $90^\circ$ as well. This means $\angle B + \angle C = 90^\circ$.

The tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side. For angles $\angle B$ and $\angle C$, we have:

$$ \tan B = \frac{\text{opposite side to } B}{\text{adjacent side to } B} $$
$$ \tan C = \frac{\text{opposite side to } C}{\text{adjacent side to } C} $$

Since $\angle B + \angle C = 90^\circ$, $\angle C$ is the complement of $\angle B$. In trigonometry, the tangent of complementary angles has a specific relationship:

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

Thus, $\tan C = \cot B$. Therefore, the product of $\tan B$ and $\tan C$ is:

$$ \tan B \times \tan C = \tan B \times \cot B = \tan B \times \frac{1}{\tan B} = 1 $$

This confirms that the value of $\tan B \times \tan C$ is indeed 1, which supports Option 3 as the correct answer.

Now, let's explain why the other options are incorrect:

1. **Option 1: -1** - The product of $\tan B$ and $\tan C$ cannot be negative because both angles are acute angles in a right triangle, and the tangent of acute angles is positive.

2. **Option 2: 0** - The product $\tan B \times \tan C$ cannot be zero unless one of the tangents is zero. However, in a right triangle, neither $\angle B$ nor $\angle C$ can be $0^\circ$ or $90^\circ$, so their tangents cannot be zero.

3. **Option 4: $\infty$** - The product $\tan B \times \tan C$ cannot be infinite because both $\tan B$ and $\tan C$ are finite positive values for acute angles in a right triangle.

Therefore, the correct answer is indeed Option 3, as the product of $\tan B$ and $\tan C$ is 1.