Find the value:
$\frac{1-tan^245}{1 + tan^2 45}$

Q: Find the value: $\frac{1-tan^245}{1 + tan^2 45}$

To solve the problem, we need to evaluate the expression $\frac{1-\tan^2 45}{1 + \tan^2 45}$. First, let's recall the value of $\tan 45^\circ$. The tangent of 45 degrees is 1, i.e., $\tan 45^\circ = 1$. Now, substitute $\tan 45^\circ = 1$ into the expression: \[ \tan^2 45 = (1)^2 = 1 \] Substitute this into the expression: \[ \frac{1 - \tan^2 45}{1 + \tan^2 45} = \frac{1 - 1}{1 + 1} = \frac{0}{2} = 0 \] Thus, the value of the expression is 0, which corresponds to Option 4. ### Explanation for Other Options: 1. **Option 1: ∞** - The expression $\frac{0}{2}$ evaluates to 0, not infinity. Infinity would occur if the denominator were zero and the numerator non-zero, which is not the case here. 2. **Option 2: 1** - For the expression to be 1, the numerator and the denominator would need to be equal. Here, the numerator is 0 and the denominator is 2, so the expression evaluates to 0, not 1. 3. **Option 3: -1** - For the expression to be -1, the numerator would need to be the negative of the denominator. Here, the numerator is 0 and the denominator is 2, so the expression evaluates to 0, not -1. Therefore, the correct answer is indeed Option 4, as the expression evaluates to 0.

Question

Find the value: 
$\frac{1-tan^245}{1 + tan^2 45}$

Accepted Answer

To solve the problem, we need to evaluate the expression $\frac{1-\tan^2 45}{1 + \tan^2 45}$.

First, let's recall the value of $\tan 45^\circ$. The tangent of 45 degrees is 1, i.e., $\tan 45^\circ = 1$.

Now, substitute $\tan 45^\circ = 1$ into the expression:

$$
\tan^2 45 = (1)^2 = 1
$$

Substitute this into the expression:

$$
\frac{1 - \tan^2 45}{1 + \tan^2 45} = \frac{1 - 1}{1 + 1} = \frac{0}{2} = 0
$$

Thus, the value of the expression is 0, which corresponds to Option 4.

### Explanation for Other Options:

1. **Option 1: ∞**
   - The expression $\frac{0}{2}$ evaluates to 0, not infinity. Infinity would occur if the denominator were zero and the numerator non-zero, which is not the case here.

2. **Option 2: 1**
   - For the expression to be 1, the numerator and the denominator would need to be equal. Here, the numerator is 0 and the denominator is 2, so the expression evaluates to 0, not 1.

3. **Option 3: -1**
   - For the expression to be -1, the numerator would need to be the negative of the denominator. Here, the numerator is 0 and the denominator is 2, so the expression evaluates to 0, not -1.

Therefore, the correct answer is indeed Option 4, as the expression evaluates to 0.