Sure, let's break this problem down step by step.
Step 1:
We start with the given expression 4 cot(θ) / (cos(θ) * csc(θ))
In order to simplify this to a constant value, it's helpful to start by understanding the definitions of the trigonometric functions involved.
Cotangent (cot) is the reciprocal of the tangent (tan), and can be written as cot(θ) = 1/tan(θ).
Cosine (cos) is a measure of the ratio of the adjacent side to the hypotenuse in a right triangle.
Cosecant (csc) is the reciprocal of the sine (sin), and can be written as csc(θ) = 1/sin(θ).
Step 2:
Now that we know these relationships, we can substitute them into our original expression.
That changes our expression to: 4 * (1/tan(θ)) / ((cos(θ) * (1/sin(θ)))
Step 3:
This simplifies to: 4/ (tan(θ) * cos(θ) * sin(θ))
Remembering that sin(θ) / cos(θ) = tan(θ), we can further simplify this to: 4/ (tan^2(θ))
Step 4:
Finally, considering that tan^2(θ) = 1 for all θ, we find that the simplified form of the expression equals 4/1
Hence, our final simplified form is equal to 4.
So, the simplified form of the expression 4 cot θ/ cos θ cscθ is 4.