Help me solve these trig identity problems!?

Question

Okay, now I need anybody who can help me with one of these problems. I have a trigonometry test coming up soon, and I don't know how to solve these problems on my review. So, if anyone can show me how to do at least one or all of these three problems, it would be greatly appreciated. Here they are:

1. tan x sin x
---------- - ----------------- = sec x + cos x
csc x - cot x csc x + cot x

2. tan^3 x + sin x sec x - sin x cos x
------------------------------------------- = sin x + tan x sec x
sec x - cos x

3. cos x - cos 2x x
------------------- = tan -----
sin x + sin2x 2

If anyone can give me some kind of advice, or at least a good explanation, ten points will be guaranteed.

For some reason this got mixed up. The first problem is tangent over csc x - cot x, minus sin x over csc x + cot x equals sec x + cos x. The second one seems alright, just that the numerator is sitting overtop the answer. The third one is cos x - cos 2x over sin x + sin2x. This should equal tan x/2. Sorry for the confusion.

Answer 1

1.) tan x / (csc x - cot x) - sin x / (csc x + cot x) = sec x + cos x

Trig identity: csc^2 x - cot^2 x = 1

So, use difference of 2 squares (a + b)(a - b) = a^2 - b^2:
tan x / (csc x - cot x) * (csc x + cot x) / (csc x + cot x) - sin x / (csc x + cot x) * (csc x - cot x) / (csc x - cot x)

tan x (csc x + cot x) / (csc^2 x - cot^2 x) - sin x (csc x - cot x) / (csc^2 x - cot^2 x)

tan x (csc x + cot x) - sin x (csc x - cot x) = sec x + cos x

Distribute parenthesis and Convert identities:
(sin x / cos x) (sin^-1 x) + (tan x * tan^-1 x) - (sin x * sin^-1 x) - (sin x * cos x / sin x)
1 / cos x + 1 - 1 - cos x
= sec x + cos x (solution!)

2.) (tan^3 x + sin x sec x - sin x cos x) / (sec x - cos x) = sin x + tan x sec x

Multiply denominator to other side:
(tan^3 x + sin x sec x - sin x cos x) = (sec x - cos x) * (sin x + tan x sec x)
= tan x - sin x cos x + tan x sec^2 x - tan x
= -sin x cos x + tan x sec^2 x

Notice the -sin x cos x on both sides: we can cancel that.
tan^3 x + sin x sec x - sin x cos x = -sin x cos x + tan x sec^2 x
tan^3 x + sin x sec x = tan x sec^2 x
(sin^3 x sec^3 x + sin x sec x) / sec x = tan x sec^2x / sec x
sin^3 x sec^2 x + sin x = tan x sec x

(sin^3 x sec^2 x + sin x) / sin x = sin x sec^2 x / sin x
sin^2 x sec^2 x + 1 = sec ^2 x
tan^2 x + 1 = sec^2 x (this is a trig identity...solution!)

3 is probably done in a similar manner using the base trig identities.

Answer 2

a million/ cos^2 x - cosx - 2 = 0 ---> (cosx+a million)(cosx-2) = 0 ---> cosx = -a million --->x = pi 2/ sin2x = (sqrt3)/2 ---> 2x = pi/3 + k2pi or 2pi/3 + k2pi , with ok = 0 , a million, 2 , ... similar for 3/4