Problems with trig?

Question

Hi.
I have two problems that I need help with, please.
1) I need to classify the following equation as an identity, conditional equation or contradiction: sin x = tan x cos x
2) I need to convert the following equation to polar form:
2x - 2y = 1
Could someone please walk me through the solutions step by step and explain each step so I can understand how to do this.
Thank you.

Answer 1

1) Identity because tanx cosx = (sinx/cosx)cosx = sinx if and only if cosx ≠ 0.
2) 2r(cosθ - sinθ) = 1 because x = rcosθ, y = rsinθ

Answer 2

1. It depends on whether or not it is ok to reduce the equation. Sometimes they don't want you to for some reason or other. Actually in this case it doesn't matter, each will give the same answer. So:
tan x = sin x / cos x
tan x cos x = (sin x / cos x)*cos x = sin x
so sin x = sin x and it is an identity

2. x and y are the coordinates of a point
The distance "r" the point is from (0,0) is SQRT(x^2 + y^2)
If T is the angle the line from (0,0) to (x,y) makes with the X-axis then you can write x and y as:
x = r*cos(T) and y = r*sin(T)

So 2x - 2y = 1 = 2r (cos(T) - sin(T))