Answer :
To demonstrate that [tex]\(\arctan(-x) = \pi - \arctan(x)\)[/tex], let's proceed with a detailed, step-by-step solution.
### Step-by-Step Solution:
1. Review of Arctangent Function:
The [tex]\(\arctan(x)\)[/tex] function, also known as the inverse tangent function, is the angle [tex]\(\theta\)[/tex] such that [tex]\(\tan(\theta) = x\)[/tex]. The range of [tex]\(\arctan(x)\)[/tex] is typically [tex]\((- \frac{\pi}{2}, \frac{\pi}{2})\)[/tex].
2. Define the Identity:
We need to show that [tex]\(\arctan(-x) = \pi - \arctan(x)\)[/tex].
3. Let [tex]\(y = \arctan(x)\)[/tex]:
This implies that [tex]\(\tan(y) = x\)[/tex]. Therefore, [tex]\(y\)[/tex] is the angle whose tangent is [tex]\(x\)[/tex].
4. Consider [tex]\(\arctan(-x)\)[/tex]:
Let [tex]\(z = \arctan(-x)\)[/tex]. This implies that [tex]\(\tan(z) = -x\)[/tex]. Therefore, [tex]\(z\)[/tex] is the angle whose tangent is [tex]\(-x\)[/tex].
5. Use Properties of the Tangent Function:
The tangent function is an odd function, meaning [tex]\(\tan(-\theta) = -\tan(\theta)\)[/tex].
6. Express in Terms of Known Angles:
If [tex]\(y = \arctan(x)\)[/tex], then [tex]\(\tan(y) = x\)[/tex]. We seek an angle such that its tangent is [tex]\(-x\)[/tex]. By the odd function property:
[tex]\[ \tan(-y) = -\tan(y) = -x \][/tex]
7. Find the Relationship:
-[tex]\(y\)[/tex] is an angle whose tangent is [tex]\(-x\)[/tex]. Therefore:
[tex]\[ z = -y \][/tex]
Since we defined [tex]\(z = \arctan(-x)\)[/tex], then we have:
[tex]\[ \arctan(-x) = -\arctan(x) \][/tex]
8. Adjust to Preferable Range:
To ensure the expression fits within the typical principal value range [tex]\( (-\frac{\pi}{2}, \frac{\pi}{2}) \)[/tex], note that:
[tex]\[ -\arctan(x) \equiv \pi - \arctan(x) \][/tex]
if [tex]\(\arctan(x)\)[/tex] lies in the range [tex]\((0, \frac{\pi}{2})\)[/tex].
This equivalence is because:
[tex]\[ \pi - (\pi - \theta) = \theta \][/tex]
### Conclusion:
Based on these properties and transformations, we have shown that:
[tex]\[ \arctan(-x) = \pi - \arctan(x). \][/tex]
This demonstrates the identity [tex]\(\arctan(-x) = \pi - \arctan(x)\)[/tex] as required.
### Step-by-Step Solution:
1. Review of Arctangent Function:
The [tex]\(\arctan(x)\)[/tex] function, also known as the inverse tangent function, is the angle [tex]\(\theta\)[/tex] such that [tex]\(\tan(\theta) = x\)[/tex]. The range of [tex]\(\arctan(x)\)[/tex] is typically [tex]\((- \frac{\pi}{2}, \frac{\pi}{2})\)[/tex].
2. Define the Identity:
We need to show that [tex]\(\arctan(-x) = \pi - \arctan(x)\)[/tex].
3. Let [tex]\(y = \arctan(x)\)[/tex]:
This implies that [tex]\(\tan(y) = x\)[/tex]. Therefore, [tex]\(y\)[/tex] is the angle whose tangent is [tex]\(x\)[/tex].
4. Consider [tex]\(\arctan(-x)\)[/tex]:
Let [tex]\(z = \arctan(-x)\)[/tex]. This implies that [tex]\(\tan(z) = -x\)[/tex]. Therefore, [tex]\(z\)[/tex] is the angle whose tangent is [tex]\(-x\)[/tex].
5. Use Properties of the Tangent Function:
The tangent function is an odd function, meaning [tex]\(\tan(-\theta) = -\tan(\theta)\)[/tex].
6. Express in Terms of Known Angles:
If [tex]\(y = \arctan(x)\)[/tex], then [tex]\(\tan(y) = x\)[/tex]. We seek an angle such that its tangent is [tex]\(-x\)[/tex]. By the odd function property:
[tex]\[ \tan(-y) = -\tan(y) = -x \][/tex]
7. Find the Relationship:
-[tex]\(y\)[/tex] is an angle whose tangent is [tex]\(-x\)[/tex]. Therefore:
[tex]\[ z = -y \][/tex]
Since we defined [tex]\(z = \arctan(-x)\)[/tex], then we have:
[tex]\[ \arctan(-x) = -\arctan(x) \][/tex]
8. Adjust to Preferable Range:
To ensure the expression fits within the typical principal value range [tex]\( (-\frac{\pi}{2}, \frac{\pi}{2}) \)[/tex], note that:
[tex]\[ -\arctan(x) \equiv \pi - \arctan(x) \][/tex]
if [tex]\(\arctan(x)\)[/tex] lies in the range [tex]\((0, \frac{\pi}{2})\)[/tex].
This equivalence is because:
[tex]\[ \pi - (\pi - \theta) = \theta \][/tex]
### Conclusion:
Based on these properties and transformations, we have shown that:
[tex]\[ \arctan(-x) = \pi - \arctan(x). \][/tex]
This demonstrates the identity [tex]\(\arctan(-x) = \pi - \arctan(x)\)[/tex] as required.
