5 Questions
How many solutions does the equation $(\cot^{-1}x)^2 - 3(\cot^{-1}x) + 2 > 0$ have?
$(\cot 1, \infty)$
What is the value of $\cos\left(\cos^{-1}\left(-\frac{1}{2}\right) + \sin\left(\sin^{-1}\left(-\frac{1}{2}\right)\right)\right)$?
0
If $1 < x < 2$, then the number of solutions of the equation $\tan^{-1}(x - 1) + \tan^{-1}x + \tan^{-1}(x + 1) = \tan^{-1}3x$ is/are:
1
What is the complete solution set of $\tan(\sin x) > 1$?
$\left(-\frac{\pi}{6}, -2\right) \cup \left(\frac{\pi}{6}, 2\right)$
If $\sin^{-1}\left(\frac{x-1}{2}\right) + \cos^{-1}\left(\frac{x+1}{2}\right) = \frac{\pi}{4}$, then what is $x(x+1) + \sin^{-1}x$?
$\frac{7}{16}$
Study Notes
Trigonometric Equations and Inequalities
Equation Solving
- The equation $(\cot^{-1}x)^2 - 3(\cot^{-1}x) + 2 > 0$ has a certain number of solutions.
- The value of $\cos\left(\cos^{-1}\left(-\frac{1}{2}\right) + \sin\left(\sin^{-1}\left(-\frac{1}{2}\right)\right)\right)$ is a specific number.
Inverse Trigonometric Functions
- The number of solutions of the equation $\tan^{-1}(x - 1) + \tan^{-1}x + \tan^{-1}(x + 1) = \tan^{-1}3x$ is dependent on the condition $1 < x < 2$.
Trigonometric Inequalities
- The complete solution set of the inequality $\tan(\sin x) > 1$ is a specific set of values.
Compositions of Inverse Trigonometric Functions
- The equation $\sin^{-1}\left(\frac{x-1}{2}\right) + \cos^{-1}\left(\frac{x+1}{2}\right) = \frac{\pi}{4}$ is related to the expression $x(x+1) + \sin^{-1}x$.
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