Product of a Derivative and its Inverse

[EconStudent]

Homework Statement

For the functions:

x = r*cos(θ)
y = r*sin(θ)

Calculate dθ/dX * dX/dθ (considering θ as a function of x, y) and simplify.

The Attempt at a Solution

I believe this should always be 1, by definition (as the product of a derivative and its inverse). However, I don't know how to show this.

Solving the first function for θ, I get cos^-1(x/r). The derivative of this is a relatively messy expression 1 over the product of r and a square root, and its being multiplied by something relatively simple (-r*sin(θ)). Why is it that this would simplify to 1?

I get that the negatives and the r's will cancel. So it simplifies to:

sin(θ) / √(1 - x²/r²)

But I can't get any further nor am I satisfied that this expression is or is not equal to 1. Thanks in advance.

 

[Dick]

Homework Statement

For the functions:

x = r*cos(θ)
y = r*sin(θ)

Calculate dθ/dX * dX/dθ (considering θ as a function of x, y) and simplify.

The Attempt at a Solution

I believe this should always be 1, by definition (as the product of a derivative and its inverse). However, I don't know how to show this.

Solving the first function for θ, I get cos^-1(x/r). The derivative of this is a relatively messy expression 1 over the product of r and a square root, and its being multiplied by something relatively simple (-r*sin(θ)). Why is it that this would simplify to 1?

I get that the negatives and the r's will cancel. So it simplifies to:

sin(θ) / √(1 - x²/r²)

But I can't get any further nor am I satisfied that this expression is or is not equal to 1. Thanks in advance.

Use that x/r=cos(theta). What does that make 1-(x/r)^2?

 

[EconStudent]

That's so clever, thank you. Substituting for x and squaring the top and bottom, we get sin²/1-cos² which is just sin²/sin².

 

[Mark44]

That's so clever, thank you. Substituting for x and squaring the top and bottom, we get sin²/1-cos² which is just sin²/sin².

When you write fractional expressions in a line, and there are multiple terms in the top or bottom, USE PARENTHESES!

What you wrote would be reasonably interpreted as
$$\frac{sin^2(x)}{1} -cos^2(x)$$