can any one prove this identity?

Question

(1/2)(sin x)[(sin x)/(1-cos x) + (1-cos x)/(sin x)] = 1

Answer 1

LS= (1/2)(sin x)[(sin x)/(1-cos x) + (1-cos x)/(sin x)]

=1/[2(1-cosx)] * [(sin x)^2 + (1-cos x)^2]
=1/[2(1-cosx)] * [1-(cos x)^2 + 1-2cosx +(cos x)^2]
=1/[2(1-cosx)] *(2+2cosx)
=1
=RS

Answer 2

I would start by getting a common denominator for the expression on brackets.

sin x / (1-cos x) * (sin x)/(sin x) = sin^2 x /[sin x (1 - cos x)]

(1 - cos x) / sin x * (1 - cos x)/(1- cos x) = (1 - cos x)^2/[sin x (1 - cos x)]

So, in the brackets, we get:

(1 - 2 cos x + cos^2 x + sin^2 x) /[sin x (1 - cos x)] (I expanded (1 - cos x)^2 to save time)

We know that sin^2 x + cos^2 x = 1, so we can make that substitution.

1 - 2 cos x + 1 = 2 - 2 cos x

So now our part in brackets is:

2(1 - cos x)/[sin x(1 - cos x)] = 2/(sin x)

Now that we know how simple the mess in the brackets really is, we can substitute it back into our equation:

1/2 (sin x) 2/(sin x) = 1

Notice how we have the number 2 on both sides of the fraction and sin x on both sides. Therefore, both cancel each other out, leaving us with 1 = 1, which is a true statement.

Answer 3

the devil, he does a lot of sins.