Verify the identity:
cos x -cox x/1-tan x = sin x cos x/sin x -cos x
To verify the given identity:
cos x - cox x / 1 - tan x = sin x cos x / sin x - cos x
First, let's simplify the left side of the identity:
cos x - cox x / 1 - tan x
We can rewrite cox x as 1 / tan x (using the reciprocal identity for tangent):
cos x - 1 / tan x / 1 - tan x
To simplify further, we can multiply the numerator and denominator of the fraction by tan x:
(cos x - 1 / tan x) * (tan x / 1 - tan x)
Now, using the distributive property, we can expand the numerator:
= cos x * tan x - 1 * tan x / 1 - tan x
Simplifying the numerator, we get:
= sin x - tan x / 1 - tan x
Now let's simplify the right side of the identity:
sin x cos x / sin x - cos x
We can factor out sin x from the numerator and cos x from the denominator:
(sin x * cos x) / (sin x - cos x)
Now we can simplify the expression:
= sin x / 1 - cos x / 1
= sin x / 1 - cos x
So, the right side of the identity simplifies to sin x / (1 - cos x).
Comparing this with the left side, we can see that they are equal:
(sin x - tan x) / (1 - tan x) = sin x / (1 - cos x)
Thus, the original identity is verified.