Prove the following to be an identity
(cotx + cscx)*(1-cosx)=sin x
(cos/sin + 1/sin)(1-cos)
=(cos/sin+1/sin)(1)+(cos/sin+1/sin)(-cos)
= cos/sin+1/sin-cos^2/sin-cos/sin
= (1/sin )(1-cos^2) but 1-cos^2=sin^2
so
sin^2/sin
= sin
sin isnt an identity to begin with
and i got sin^2 x + cos^2 x = 1
the ansswer says its what i got
The right hand side says sin x
You must make the left hand side come out sin x
That is what I did
Yes, I also used sin^2 + cos^2 = 1
in saying
1 - cos^2 = sin^2
To prove that the expression (cotx + cscx)*(1-cosx) is an identity, we need to simplify both sides of the equation and show that they are equal.
Let's start with the left side of the equation and simplify it step by step:
(cotx + cscx)*(1 - cosx)
Using the reciprocal identities, we can rewrite cotx as 1/tanx and cscx as 1/sinx:
(1/tanx + 1/sinx)*(1 - cosx)
Next, let's find a common denominator for the fractions:
((sinx + tanx)/sinx)*(1 - cosx)
Now, let's simplify the expression by multiplying the numerators:
(sin x + tan x)(1 - cos x)/sin x
Using the distributive property, we have:
(sin x - sin x cos x + tan x - tan x cos x)/sin x
Now, let's factor out a common term:
sin x(1 - cos x) + tan x(1 - cos x)/sin x
Using the fact that tan x = sin x / cos x, we can simplify further:
sin x(1 - cos x) + sin x / cos x * (1 - cos x)
Now, let's find a common denominator for the two terms:
(sin x cos x - sin x cos^2 x + sin x - sin x cos x) / cos x
Combining like terms:
sin x - sin x cos^2 x / cos x
We can simplify sin x / cos x to get:
sin x - sin x cos x
Using the fact that sin x / cos x = tan x:
tan x - sin x cos x
Therefore, the left side of the equation simplifies to tan x - sin x cos x.
Now, let's simplify the right side of the equation, which is sin x:
Thus, the right side is sin x.
Since the left side (tan x - sin x cos x) is equal to the right side (sin x), we have shown that the expression (cotx + cscx)*(1-cosx) is an identity.
Therefore, (cotx + cscx)*(1-cosx) = sin x is true for all values of x.