x ∈ R
Understanding the Domain of \(\rm tan^{-1}x + cot^{-1}x = \frac{\pi}{2}\)
The question asks for the values of \(x\) for which the identity \(\rm tan^{-1}x + cot^{-1}x = \frac{\pi}{2}\) holds true. To understand this, we need to consider the domains of the individual inverse trigonometric functions involved.
Domains of Inverse Trigonometric Functions
- The domain of the tangent inverse function, \(\rm tan^{-1}x\), is the set of all real numbers. This means that \(\rm tan^{-1}x\) is defined for any real value of \(x\). We can write this as \(x \in \rm R\).
- The domain of the cotangent inverse function, \(\rm cot^{-1}x\), is also the set of all real numbers. This means that \(\rm cot^{-1}x\) is defined for any real value of \(x\). We can write this as \(x \in \rm R\).
Condition for the Identity to Hold
For the expression \(\rm tan^{-1}x + cot^{-1}x\) to be defined, both \(\rm tan^{-1}x\) and \(\rm cot^{-1}x\) must be defined. Therefore, \(x\) must belong to the intersection of their domains.
Domain of \(\rm tan^{-1}x\) is \(\rm R\).
Domain of \(\rm cot^{-1}x\) is \(\rm R\).
The intersection of the domains is \(\rm R \cap R = R\).
So, the sum \(\rm tan^{-1}x + cot^{-1}x\) is defined for all real numbers, i.e., for \(x \in \rm R\).
The Standard Identity
The identity \(\rm tan^{-1}x + cot^{-1}x = \frac{\pi}{2}\) is a fundamental property of inverse trigonometric functions. This identity is known to hold true for all values of \(x\) within the common domain of \(\rm tan^{-1}x\) and \(\rm cot^{-1}x\).
Since the common domain is the set of all real numbers (\(x \in \rm R\)), the identity \(\rm tan^{-1}x + cot^{-1}x = \frac{\pi}{2}\) is valid for all \(x \in \rm R\).
Analyzing the Options
Let's look at the given options:
- Option 1: \(x \in \rm R\). This matches our conclusion that the identity holds for all real numbers.
- Option 2: \(x \in \rm R - (-1, 1)\) only. This restricts the domain unnecessarily.
- Option 3: \(x \in \rm R - \{0\}\) only. This also restricts the domain unnecessarily.
- Option 4: \(x \in \rm R - [-1, 1]\) only. This restricts the domain unnecessarily.
Based on the domains of \(\rm tan^{-1}x\) and \(\rm cot^{-1}x\) and the standard identity, the equation \(\rm tan^{-1}x + cot^{-1}x = \frac{\pi}{2}\) holds for all real numbers \(x\).
Conclusion
The identity \(\rm tan^{-1}x + cot^{-1}x = \frac{\pi}{2}\) holds for all \(x \in \rm R\).
Revision Table: Inverse Trigonometric Function Domains
| Function | Domain | Range |
|---|---|---|
| \(\rm sin^{-1}x\) | \([-1, 1]\) | \([-\frac{\pi}{2}, \frac{\pi}{2}]\) |
| \(\rm cos^{-1}x\) | \([-1, 1]\) | \([0, \pi]\) |
| \(\rm tan^{-1}x\) | \(\rm R\) | \((-\frac{\pi}{2}, \frac{\pi}{2})\) |
| \(\rm cot^{-1}x\) | \(\rm R\) | \((0, \pi)\) |
| \(\rm sec^{-1}x\) | \(\rm (-\infty, -1] \cup [1, \infty)\) | \([0, \pi] - \{\frac{\pi}{2}\}\) |
| \(\rm cosec^{-1}x\) | \(\rm (-\infty, -1] \cup [1, \infty)\) | \([-\frac{\pi}{2}, \frac{\pi}{2}] - \{0\}\) |
Additional Information on Inverse Trig Identities
Besides \(\rm tan^{-1}x + cot^{-1}x = \frac{\pi}{2}\), there are other important inverse trigonometric identities:
- \(\rm sin^{-1}x + cos^{-1}x = \frac{\pi}{2}\), which holds for \(x \in [-1, 1]\). Note the difference in domain compared to the \(\rm tan^{-1}x + cot^{-1}x\) identity.
- \(\rm sec^{-1}x + cosec^{-1}x = \frac{\pi}{2}\), which holds for \(x \in (-\infty, -1] \cup [1, \infty)\). Again, the domain depends on the functions involved.
It is crucial to remember the specific domain for which each inverse trigonometric identity is valid, as this determines the range of \(x\) values for which the equation holds.
Similar Questions
- If \(4 \sin^{-1} x + \cos^{-1} x = \pi\), then what is \(\sin^{-1} x + 4 \cos^{-1} x\) equal to?
- What is \(\cot^2 (\sec^{-1}2) + \tan^2 (\text{cosec}^{-1}3)\) equal to?
- Consider the following in respect of inverse circular functions :
I. \(\sin^{-1}(-x) = -\sin^{-1} x\)
II. \(\cos^{-1}(-x) = \cos^{-1} x\)
III. \(\tan^{-1}(-x) = \pi - \tan^{-1} x\)
IV. \(\cot^{-1}(-x) = \pi - \cot^{-1} x\)
How many of the above are correct ? - What is \(\tan \left[ 2 \tan^{-1} \frac{1}{2} - \frac{\pi}{4} \right]\) equal to ?
For what value of θ, where 0 < θ < \(\frac{\pi}{2}\) , does sin θ + sin θ cos θ maximum value?
What is the principal value of \({\sin ^{ - 1}}\left( {\sin \frac{{2{\rm{\pi }}}}{3}} \right)?\) ?
What is the value of cos (2 cos -1 0.8)?
Consider the following statements:
1. \({\tan ^{ - 1}}{\rm{x}} + {\tan ^{ - 1}}\left( {\frac{1}{{\rm{x}}}} \right) = {\rm{\pi }}\)
2. There exist x, y ∈ [-1, 1], where x ≠ y such that sin -1 x + cos -1 \({\rm{y}} = \frac{{\rm{\pi }}}{2}\)
Which of the above statements is/are correct?The value of \({\rm{tan}}\left( {2{{\tan }^{ - 1}}\frac{1}{5} - \frac{\pi }{4}} \right)\) is
Consider the following:
1. \({\sin ^{ - 1}}\frac{4}{5} + {\sin ^{ - 1}}\frac{3}{5} = \frac{\pi }{2}\)
2. \({\tan ^{ - 1}}\sqrt 3 + {\tan ^{ - 1}}1 = - {\tan ^{ - 1}}\left( {2 + \sqrt 3 } \right)\)
Which of the above is/are correct?
Important Questions from Inverse Trigonometric Functions
Solve for x?
sin 2x - 4 sin x + 3 = 0, 0 ≤ x ≤ \(π/2\)
What is the principal solutions of the equation \(\tan x=-\frac{1}{\sqrt{3}}\) ?
- If \(4 \sin^{-1} x + \cos^{-1} x = \pi\), then what is \(\sin^{-1} x + 4 \cos^{-1} x\) equal to?
- What is \(\cot^2 (\sec^{-1}2) + \tan^2 (\text{cosec}^{-1}3)\) equal to?
- Consider the following in respect of inverse circular functions :
I. \(\sin^{-1}(-x) = -\sin^{-1} x\)
II. \(\cos^{-1}(-x) = \cos^{-1} x\)
III. \(\tan^{-1}(-x) = \pi - \tan^{-1} x\)
IV. \(\cot^{-1}(-x) = \pi - \cot^{-1} x\)
How many of the above are correct ?
