If sec-1 p - cosec-1q = 0, where p > 0, q > 0; then what is the value of p-2 + q-2 ?
1
Understanding the Inverse Trigonometric Equation
We are given the equation \(\sec^{-1} p - \csc^{-1} q = 0\), with the conditions that \(p > 0\) and \(q > 0\). Our goal is to find the value of \(p^{-2} + q^{-2}\).
Let's start by rearranging the given equation:
\(\sec^{-1} p = \csc^{-1} q\)
Let's assume that this common value is \(\theta\). So, we have:
\(\sec^{-1} p = \theta \quad \text{and} \quad \csc^{-1} q = \theta\)
Analyzing the Domain and Range
For the inverse secant function, \(\sec^{-1} p\), the domain is \((-\infty, -1] \cup [1, \infty)\) and the principal value range is \([0, \pi/2) \cup (\pi/2, \pi]\).
For the inverse cosecant function, \(\csc^{-1} q\), the domain is \((-\infty, -1] \cup [1, \infty)\) and the principal value range is \([-\pi/2, 0) \cup (0, \pi/2]\).
Given that \(p > 0\) and \(q > 0\), and considering the domains, we must have \(p \ge 1\) and \(q \ge 1\).
The common value \(\theta\) must be in the intersection of the ranges where \(\sec^{-1}\) and \(\csc^{-1}\) are defined for positive arguments. This intersection is \((0, \pi/2]\).
If \(\theta = \pi/2\), \(\sec(\pi/2)\) is undefined. Therefore, \(\theta\) cannot be \(\pi/2\). This means \(\theta\) must be in the interval \((0, \pi/2)\).
Using Trigonometric Definitions
From the inverse relations, if \(\sec^{-1} p = \theta\), then by definition, \(p = \sec \theta\).
Similarly, if \(\csc^{-1} q = \theta\), then by definition, \(q = \csc \theta\).
Now, we can express \(\sec \theta\) and \(\csc \theta\) in terms of basic trigonometric functions, sine and cosine:
- \(p = \sec \theta = \frac{1}{\cos \theta}\)
- \(q = \csc \theta = \frac{1}{\sin \theta}\)
From these relationships, we can find \(\cos \theta\) and \(\sin \theta\) in terms of \(p\) and \(q\):
- \(\cos \theta = \frac{1}{p}\)
- \(\sin \theta = \frac{1}{q}\)
Applying a Fundamental Trigonometric Identity
We know the fundamental trigonometric identity relating sine and cosine for any angle \(\theta\):
\(\sin^2 \theta + \cos^2 \theta = 1\)
Now, substitute the expressions for \(\sin \theta\) and \(\cos \theta\) that we found:
\(\left(\frac{1}{q}\right)^2 + \left(\frac{1}{p}\right)^2 = 1\)
Squaring the terms gives:
\(\frac{1}{q^2} + \frac{1}{p^2} = 1\)
Recall that \(x^{-n} = \frac{1}{x^n}\). So, we can write \(\frac{1}{q^2}\) as \(q^{-2}\) and \(\frac{1}{p^2}\) as \(p^{-2}\).
Therefore, the equation becomes:
\(p^{-2} + q^{-2} = 1\)
This is exactly the expression we were asked to find the value of.
Conclusion
The value of \(p^{-2} + q^{-2}\) is 1.
Revision Table: Inverse Trigonometric Functions
| Function | Domain | Principal Value Range |
|---|---|---|
| \(\sec^{-1} x\) | \((-\infty, -1] \cup [1, \infty)\) | \([0, \pi/2) \cup (\pi/2, \pi]\) |
| \(\csc^{-1} x\) | \((-\infty, -1] \cup [1, \infty)\) | \([-\pi/2, 0) \cup (0, \pi/2]\) |
| \(\sin \theta\) | All real numbers | \([-1, 1]\) |
| \(\cos \theta\) | All real numbers | \([-1, 1]\) |
Additional Information: Trigonometric Identities
Trigonometric identities are equations that are true for all values of the variable for which both sides of the equation are defined. The identity \(\sin^2 \theta + \cos^2 \theta = 1\) is one of the most fundamental Pythagorean identities.
Other related Pythagorean identities derived from this one are:
- \(1 + \tan^2 \theta = \sec^2 \theta\) (Divide \(\sin^2 \theta + \cos^2 \theta = 1\) by \(\cos^2 \theta\))
- \(1 + \cot^2 \theta = \csc^2 \theta\) (Divide \(\sin^2 \theta + \cos^2 \theta = 1\) by \(\sin^2 \theta\))
These identities are crucial when simplifying trigonometric expressions or solving trigonometric equations, as seen in this problem where we used \(\sin^2 \theta + \cos^2 \theta = 1\) to relate \(p\) and \(q\) derived from \(\sec \theta\) and \(\csc \theta\). Understanding the definitions and properties of inverse trigonometric functions and fundamental identities is key to solving such problems.
Similar Questions
What is 2 cot \(\left(\frac{1}{2} \cos ^{-1} \frac{\sqrt{5}}{3}\right)\) equal to ?
The equation \({\tan ^{ - 1}}\left( {1 + {\rm{x}}} \right) + {\tan ^{ - 1}}\left( {1 - {\rm{x}}} \right) = \frac{{\rm{\pi }}}{2}\) is satisfied by
- What is tan −1 cot(cosec −1 2) equal to ?
The equation \(sin^{-1}x-cos^{-1}x=\frac{\pi}{6}\) has
What is \(\tan \left\{ 2{{\tan }^{-1}}\left( \frac{1}{3} \right) \right\}\) equal to?
What is the value of \({\sin ^{ - 1}}\frac{4}{5} + {\sec ^{ - 1}}\frac{5}{4} - \frac{\pi }{2}?\)
If \({\sin ^{ - 1}}\frac{{2p}}{{1 + p2}} - {\cos ^{ - 1}}\frac{{1 - {q^2}}}{{1 + {q^2}}} = {\tan ^{ - 1}}\frac{{2x}}{{1 - {x^2}}}\) , then what is x equal to?
Consider the following values of x:
1) 8
2) -4
3) \(\frac 16\)
4) \(- \frac{1}{4}\)
Which of the above values of x is/are the solution(s) of the equation
\({\tan ^{ - 1}}\left( {2x} \right) + {\tan ^{ - 1}}\left( {3x} \right) = \frac{\pi }{4}?{\rm{\;}}\)
What is \(\tan ^{- 1}\left( {\frac{1}{4}} \right) + {\tan ^{ - 1}}\left( {\frac{3}{5}} \right)\) equal to?
Let the equation sec x.cosec x = p have a solution, where p is a positive real number. What should be the smallest value of p?
Important Questions from Inverse Trigonometric Functions
What is 2 cot \(\left(\frac{1}{2} \cos ^{-1} \frac{\sqrt{5}}{3}\right)\) equal to ?
The principal value of sin−1\(\frac{1}{\sqrt{2}}\) is equal to which of the following?
The imaginary part of log sin (x + iy) is:
The value of \({\tan ^{ - 1}}\left( {\frac{1}{2}} \right) + {\tan ^{ - 1}}\left( {\frac{1}{3}} \right)\) is
The function \(f(x) = \sqrt {\cos (\sin x)} + {\sin ^{ - 1}}\left( {\frac{{1 + {x^2}}}{{2x}}} \right)\) is defined for
