(sin x / 1+cos x) + (1+cos x / sin x) = (2 / sin x) Use the fundamental identities to verify the following.?

Question

Show all of your work, including what calculator steps you used, if any.

Answer 1

Prove the identity.

Watch your parentheses. I assume you mean:

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

Let's start with the left hand side.

Left Hand Side = sin x / (1 + cos x) + (1 + cos x) / sin x

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

= sin x (1 - cos x) / (1 - cos²x)] + sin x (1 + cos x) / sin²x

= [sin x - (sin x)(cos x)] / sin²x + [sin x + (sin x)(cos x)] / sin²x

= [sin x - (sin x)(cos x) + sin x + (sin x)(cos x)] / sin²x

= 2sin x / sin²x = 2 / sin x = Right Hand Side