What are the steps to simplify a trig identity with multiple angles?

[phys1618]

Homework Statement

(sin 3α/sin α) - (cos 3α/cosα) =2

Homework Equations

The Attempt at a Solution

I know for sin 2 α I would put 2 sinαcosα, so for 3α, do I just put 3sinαcosα?
for cos 3α, I'm sort of clueless because there's 3 we can use for cosine,
Then after that step, I know to get both of them on the LHS to have a common denominator, which sinα cosα, please help. Thank yyou in advance!

 

[symbolipoint]

Start with the angle-sum formulas.

 

[Architeuthis Dux]

I would approach this problem by first recognizing that trig identities are mathematical equations that relate different trigonometric functions to each other. The goal of simplifying a trig identity with multiple angles is to express it in a simpler form that is easier to work with. To do this, we can follow these steps:

1. Identify and understand the given trig identity: In this case, we have (sin 3α/sin α) - (cos 3α/cosα) = 2. We need to simplify this expression by using trig identities.

2. Use known trig identities: We can use the identities sin 2α = 2sinαcosα and cos 2α = cos²α - sin²α to simplify the expression. We can also use the Pythagorean identity sin²α + cos²α = 1.

3. Convert multiple angles to single angles: In this expression, we have 3α and α as the angles. We can use the double angle identities to convert the expression to a single angle. For example, sin 3α = sin(2α + α) = sin 2α cos α + cos 2α sin α.

4. Rewrite the expression: Now, we can rewrite the expression as (2sinαcosαcos α + cos²αsin α - cos²αsin α)/sin αcos α. Simplifying this, we get (2sinαcosαcos α)/sin αcos α = 2cos α. This leaves us with (cos²αsin α - cos²αsin α)/sin αcos α = 0.

5. Simplify further: Using the Pythagorean identity, we can simplify the expression to (sin²α - cos²α)/sin αcos α = -cos 2α/sin αcos α.

6. Final step: We now have a single trig function in the expression, which is cos 2α. We can use the double angle identity cos 2α = 1 - 2sin²α to simplify the expression further. This gives us 1 - 2sin²α/sin αcos α = 1 - 2tan α.

Therefore, the simplified form of the given trig identity is 1 - 2tan α = 2. This can be verified by plugging in any value for α.

In conclusion, simplifying a trig identity with multiple