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Inverse trigonometric functions are functions that "undo" the effects of trigonometric functions. They provide a way to find the angle (or value) associated with a given trigonometric ratio. Inverse trigonometric functions are denoted by \(\sin^{-1}(x)\), \(\cos^{-1}(x)\), \(\tan^{-1}(x)\), \(\cot^{-1}(x)\), \(\sec^{-1}(x)\), and \(\csc^{-1}(x)\), representing arcsine, arccosine, arctangent, arccotangent, arcsecant, and arccosecant, respectively.

Here's a brief explanation of each inverse trigonometric function:

1. **arcsin (or \(\sin^{-1}(x)\))**: Gives the angle whose sine is \(x\), where \(x\) is between -1 and 1.

2. **arccos (or \(\cos^{-1}(x)\))**: Gives the angle whose cosine is \(x\), where \(x\) is between -1 and 1.

3. **arctan (or \(\tan^{-1}(x)\))**: Gives the angle whose tangent is \(x\).

4. **arccot (or \(\cot^{-1}(x)\))**: Gives the angle whose cotangent is \(x\).

5. **arcsec (or \(\sec^{-1}(x)\))**: Gives the angle whose secant is \(x\), where \(x \geq 1\) or \(x \leq -1\).

6. **arccsc (or \(\csc^{-1}(x)\))**: Gives the angle whose cosecant is \(x\), where \(x \geq 1\) or \(x \leq -1\).

It's important to note that the range of inverse trigonometric functions is restricted to ensure that they are single-valued and have unique inverses. The specific range depends on the convention used, but commonly accepted ranges are as follows:

- For arcsin and arccos: \(-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}\) (or \(-90^\circ \leq \theta \leq 90^\circ\)).
- For arctan: \(-\frac{\pi}{2} < \theta < \frac{\pi}{2}\) (or \(-90^\circ < \theta < 90^\circ\)).
- For arccot: \(0 < \theta < \pi\) (or \(0^\circ < \theta < 180^\circ\)).
- For arcsec and arccsc: \(0 \leq \theta < \frac{\pi}{2}\) and \(\frac{\pi}{2} < \theta \leq \pi\) (or \(0^\circ \leq \theta < 90^\circ\) and \(90^\circ < \theta \leq 180^\circ\)).

These functions are essential in solving trigonometric equations, modeling periodic phenomena, and various applications in science, engineering, and mathematics.

We know,

Let's denote \( \tan^{-1} \left( \frac{x - 1}{x - 2} \right) \) as \( \alpha \) and \( \tan^{-1} \left( \frac{x + 1}{x + 2} \right) \) as \( eta \).

Given that \( \tan^{-1} \left( \frac{x - 1}{x - 2} \right) + \tan^{-1} \left( \frac{x + 1}{x + 2} \right) = \frac{\theta}{4} \), we can use the tangent addition formula:

\[ \tan(\alpha + eta) = \frac{\tan \alpha + \tan eta}{1 - \tan \alpha \cdot \tan eta} \]

Substitute \( \tan \alpha = \frac{x - 1}{x - 2} \) and \( \tan eta = \frac{x + 1}{x + 2} \):

\[ \tan(\alpha + eta) = \frac{\frac{x - 1}{x - 2} + \frac{x + 1}{x + 2}}{1 - \frac{x - 1}{x - 2} \cdot \frac{x + 1}{x + 2}} \]

\[ \tan(\alpha + eta) = \frac{\frac{(x - 1)(x + 2) + (x + 1)(x - 2)}{(x - 2)(x + 2)}}{1 - \frac{(x - 1)(x + 1)}{(x - 2)(x + 2)}} \]

\[ \tan(\alpha + eta) = \frac{x^2 + x - 2 + x^2 - x - 2}{(x - 2)(x + 2) - (x^2 - 1)} \]

\[ \tan(\alpha + eta) = \frac{2x^2 - 4}{x^2 + 4 - x^2 + 1} \]

\[ \tan(\alpha + eta) = \frac{2x^2 - 4}{5} \]

Given that \( \tan(\alpha + eta) = \frac{\theta}{4} \), we have:

\[ \frac{2x^2 - 4}{5} = \frac{\theta}{4} \]

\[ 8x^2 - 16 = 5\theta \]

\[ 8x^2 = 5\theta + 16 \]

\[ x^2 = \frac{5\theta + 16}{8} \]

\[ x = \pm \sqrt{\frac{5\theta + 16}{8}} \]

So, the value of \( x \) depends on the value of \( \theta \).

In LaTeX code:
\[ x = \pm \sqrt{\frac{5\theta + 16}{8}} \]

To find the principal value of cos⁡−1(3−2)cos⁡−1(−32)cos−1(3−2)cos−1(−23), we'll start by finding the individual principal values of cos⁡−1(3−2)cos−1(3−2

) and cos⁡−1(−32)cos−1(−23), and then multiply them together.

1. cos⁡−1(3−2)cos−1(3−2

1. ):

Since cos⁡−1cos−1 gives an angle whose cosine is equal to the given value, we need to find an angle θθ such that cos⁡θ=3−2cosθ=3−2

.

However, the cosine function only returns values between -1 and 1. Therefore, 3−23−2

is outside the range of the cosine function, and there is no real angle θθ for which cos⁡θ=3−2cosθ=3−2. Hence, cos⁡−1(3−2)cos−1(3−2

) is undefined.

2. cos⁡−1(−32)cos−1(−23):

Similarly, −32−23 is outside the range of the cosine function, and there is no real angle ϕϕ for which cos⁡ϕ=−32cosϕ=−23. Hence, cos⁡−1(−32)cos−1(−23) is also undefined.

Therefore, the principal value of cos⁡−1(3−2)cos⁡−1(−32)cos−1(3−2

A square matrix is a matrix that has the same number of rows and columns. In other words, it is a matrix where the number of rows is equal to the number of columns.

For example, a 3×33×3 matrix and a 4×44×4 matrix are both square matrices because they have 3 rows and 3 columns, and 4 rows and 4 columns, respectively.

Square matrices are commonly encountered in various mathematical contexts, such as linear algebra, where they are used to represent linear transformations, systems of linear equations, and many other mathematical structures.