In a right angled triangle ABC, right angle is at C, if tan A = 1, then value of sin A cos A is :

Question

Accepted Answer

In the given problem, we have a right-angled triangle ABC with the right angle at C. We are given that $\tan A = 1$. We need to find the value of $\sin A \cos A$.

### Explanation:

1. **Understanding $\tan A = 1$:**
   - The tangent of angle A is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle. Therefore, $\tan A = \frac{\text{opposite}}{\text{adjacent}} = 1$.
   - This implies that the opposite side and the adjacent side are equal in length. Hence, triangle ABC is an isosceles right-angled triangle.

2. **Determining the angles:**
   - In an isosceles right-angled triangle, the two non-right angles are equal. Since the sum of angles in a triangle is $180^\circ$, and one angle is $90^\circ$, the other two angles must each be $45^\circ$.
   - Therefore, $\angle A = 45^\circ$.

3. **Calculating $\sin A$ and $\cos A$:**
   - For $\angle A = 45^\circ$, we know from trigonometric identities:
     $$
     \sin 45^\circ = \cos 45^\circ = \frac{1}{\sqrt{2}}
     $$

4. **Finding $\sin A \cos A$:**
   - Using the values of $\sin A$ and $\cos A$:
     $$
     \sin A \cos A = \left(\frac{1}{\sqrt{2}}\right) \left(\frac{1}{\sqrt{2}}\right) = \frac{1}{2}
     $$

### Justification for Option 3:

- The calculation above shows that $\sin A \cos A = \frac{1}{2}$, which matches Option 3.

### Explanation for Incorrect Options:

1. **Option 1: 1**
   - This option is incorrect because $\sin A \cos A$ for $\angle A = 45^\circ$ results in $\frac{1}{2}$, not 1.

2. **Option 2: $\frac{1}{\sqrt{2}}$**
   - This option is incorrect because $\frac{1}{\sqrt{2}}$ is the value of $\sin 45^\circ$ or $\cos 45^\circ$ individually, not their product.

3. **Option 4: 2**
   - This option is incorrect because the product of $\sin A$ and $\cos A$ for $\angle A = 45^\circ$ is $\frac{1}{2}$, not 2.

Thus, the correct answer is indeed Option 3: $\frac{1}{2}$.