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Let's go through the problems one by one.

### 1. Prove that \( \sin^{-1} \left(\frac{4}{5}\right) + \cos^{-1} \left(\frac{3}{5}\right) = \frac{\pi}{2} \).

**Solution:**

Given the equation:
\[
\sin^{-1} \left(\frac{4}{5}\right) + \cos^{-1} \left(\frac{3}{5}\right)
\]

We know that:
\[
\sin^{-1} x + \cos^{-1} x = \frac{\pi}{2}
\]

So, if \( \sin^{-1} \left(\frac{4}{5}\right) = \theta \), then \( \sin(\theta) = \frac{4}{5} \). Therefore, by the Pythagorean identity \( \cos^2 \theta = 1 - \sin^2 \theta \), we have:
\[
\cos(\theta) = \sqrt{1 - \left(\frac{4}{5}\right)^2} = \frac{3}{5}
\]

This means:
\[
\cos^{-1} \left(\frac{3}{5}\right) = \frac{\pi}{2} - \sin^{-1} \left(\frac{4}{5}\right)
\]
Thus:
\[
\sin^{-1} \left(\frac{4}{5}\right) + \cos^{-1} \left(\frac{3}{5}\right) = \frac{\pi}{2}
\]
Proven.

### 2. Find the angle between \(5i\) and \(2 + 2i\).

**Solution:**

The angle of a complex number \(z = x + yi\) with the positive real axis is given by \( \theta = \arg(z) = \tan^{-1} \left(\frac{y}{x}\right) \).

For \(5i\):
\[
\theta_1 = \tan^{-1} \left(\frac{5}{0}\right) = \frac{\pi}{2}
\]

For \(2 + 2i\):
\[
\theta_2 = \tan^{-1} \left(\frac{2}{2}\right) = \tan^{-1}(1) = \frac{\pi}{4}
\]

The angle between them is:
\[
\theta = \frac{\pi}{2} - \frac{\pi}{4} = \frac{\pi}{4}
\]

### 3. Solve for \(x\) in \(15 - 6\left(\frac{12 - x}{3}\right) = 4x\).

**Solution:**

Start by simplifying the equation:
\[
15 - 6\left(\frac{12 - x}{3}\right) = 4x
\]
\[
15 - 6\left(4 - \frac{x}{3}\right) = 4x
\]
\[
15 - (24 - 2x) = 4x
\]
\[
15 - 24 + 2x = 4x
\]
\[
-9 + 2x = 4x
\]
\[
-9 = 2x
\]
\[
x = -\frac{9}{2}
\]

### 4. For what values of \(x\) is the matrix \(A = \begin{pmatrix} 1 & 2

The set of mathematical problems in the image, including trigonometry, matrices, and synthetic division.

This set of advanced math problems covers key topics such as trigonometry, matrices, and synthetic division. Learn how to solve these problems with step-by-step solutions, suitable for students in Grades 10-12.

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