Solve the equation `cos(tan^(-1)x) = sin(cot^(-1)(3/4))`.

Explanation

To solve the equation \( \cos(\tan^{-1} x) = \sin(\cot^{-1}(3/4)) \), we will follow these steps:

### Step 1: Solve the Left-Hand Side (LHS)
We start with the LHS, which is \( \cos(\tan^{-1} x) \).

Let \( \tan^{-1} x = \alpha \). Then, by the definition of the inverse tangent function, we have:
\[ \tan \alpha = x \]

### Step 2: Construct a Right Triangle
From the definition of tangent, we can construct a right triangle where:
- The opposite side (perpendicular) is \( x \)
- The adjacent side (base) is \( 1 \)

Using the Pythagorean theorem, we can find the hypotenuse:
\[ \text{Hypotenuse} = \sqrt{x^2 + 1^2} = \sqrt{x^2 + 1} \]

### Step 3: Find \( \cos \alpha \)
Now, we can find \( \cos \alpha \):
\[ \cos \alpha = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{1}{\sqrt{x^2 + 1}} \]

Thus, the LHS becomes:
\[ \cos(\tan^{-1} x) = \frac{1}{\sqrt{x^2 + 1}} \]

### Step 4: Solve the Right-Hand Side (RHS)
Now, we evaluate the RHS, which is \( \sin(\cot^{-1}(3/4)) \).

Let \( \cot^{-1}(3/4) = \theta \). Then, by the definition of the inverse cotangent function:
\[ \cot \theta = \frac{3}{4} \]

### Step 5: Construct a Right Triangle for RHS
From the definition of cotangent, we can construct a right triangle where:
- The adjacent side (base) is \( 3 \)
- The opposite side (perpendicular) is \( 4 \)

Using the Pythagorean theorem, we can find the hypotenuse:
\[ \text{Hypotenuse} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 6: Find \( \sin \theta \)
Now, we can find \( \sin \theta \):
\[ \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{4}{5} \]

Thus, the RHS becomes:
\[ \sin(\cot^{-1}(3/4)) = \frac{4}{5} \]

### Step 7: Set LHS Equal to RHS
Now we equate the two sides:
\[ \frac{1}{\sqrt{x^2 + 1}} = \frac{4}{5} \]

### Step 8: Cross-Multiply
Cross-multiplying gives:
\[ 5 = 4\sqrt{x^2 + 1} \]

### Step 9: Square Both Sides
Squaring both sides results in:
\[ 25 = 16(x^2 + 1) \]

### Step 10: Simplify the Equation
Expanding and simplifying:
\[ 25 = 16x^2 + 16 \]
\[ 16x^2 = 25 - 16 \]
\[ 16x^2 = 9 \]

### Step 11: Solve for \( x \)
Dividing both sides by 16:
\[ x^2 = \frac{9}{16} \]
Taking the square root:
\[ x = \pm \frac{3}{4} \]

### Final Answer
Thus, the solution to the equation is:
\[ x = \frac{3}{4} \quad \text{or} \quad x = -\frac{3}{4} \]