Question

Pls derive this . r=cosx cot x

Answer 1

The student's question about the trigonometric identity r=cos(x)cot(x) was addressed by using basic trigonometric relationships and identities to derive a simplified form r=cos²(x)/sin(x) in terms of sine and cosine.

Step-by-step explanation:

The question involves deriving a trigonometric identity involving a polar coordinate expression r = cos(x) cot(x). Firstly, let's remember that in polar coordinates, the position (x, y) can be expressed using the relationships x = r cos(θ) and y = r sin(θ). Additionally, in a right triangle, cos(θ) is defined as the adjacent over hypotenuse, while sin(θ) is defined as the opposite over hypotenuse, and tan(θ) is defined as the opposite over adjacent (sin(θ)/cos(θ)). Given these definitions, cot(θ) is simply the reciprocal of tan(θ), or cos(θ)/sin(θ).

Let's apply this to the given expression r = cos(x) cot(x).

Since

cot(x) = cos(x)/sin(x),

the expression simplifies to

r = (cos(x) × cos(x))/sin(x),

or r = cos^2(x)/sin(x).

In summary, we have taken the original expression and, using trigonometric definitions and identities, derived a simplified form of the expression based on cosines and sines.

Answer 2

You must use product rule:

(dr/dx) = cos(x) * dx/dr(cot(x)) + (dx/dr)cos(x) * cot(x)

r' = cos(x) * -csc²(x) + -sin(x) * cot(x)

If simplification is needed:

r' = -cos(x) *
`(1)/(sin^(2)(x))` - sin(x) *
`(cos(x))/(sin(x))`

r' =
`(-cos(x))/(sin^(2)(x))` - cos(x)

There are other forms. Good luck!