Introduction:

In this problem, we need to solve the equation cos(tan inverse x) = sin(cot inverse 3/4) for x.

Step 1: Finding the value of cot inverse 3/4

We know that cot inverse is the inverse of cotangent function. Therefore, we can use the identity cot inverse x = tan inverse (1/x) to find the value of cot inverse 3/4 as follows:
cot inverse 3/4 = tan inverse (4/3)

Step 2: Finding the value of sin(cot inverse 3/4)

We know that sin is the opposite side/hypotenuse in a right-angled triangle. Therefore, we can draw a right-angled triangle with the adjacent side as 3 and the opposite side as 4 to find the value of sin(cot inverse 3/4) as follows:
sin(cot inverse 3/4) = 4/5

Step 3: Finding the value of tan inverse x

We know that tan inverse is the inverse of tangent function. Therefore, we can use the identity tan inverse x = cot inverse (1/x) to find the value of tan inverse x as follows:
tan inverse x = cot inverse (1/x)

Step 4: Finding the value of cos(tan inverse x)

We know that cos is the adjacent side/hypotenuse in a right-angled triangle. Therefore, we can draw a right-angled triangle with the adjacent side as 1 and the opposite side as x to find the value of cos(tan inverse x) as follows:
cos(tan inverse x) = 1/√(1+x²)

Step 5: Equating the two sides of the equation

Now, we can equate the two sides of the equation cos(tan inverse x) = sin(cot inverse 3/4) and solve for x as follows:
1/√(1+x²) = 4/5
Squaring both sides, we get:
1/(1+x²) = 16/25
Solving for x, we get:
x = ±3/4

Conclusion:

Therefore, the solution to the equation cos(tan inverse x) = sin(cot inverse 3/4) is x = ±3/4.