what conjecture could be made and proven from the following?

sin(3x)/sin(x) - cos(3x)/cos(x)

I put it on my calculator and based on the graph I say that as it crosses the y axis the values get larger.

To make a conjecture and prove it based on the given expression, let's first simplify it:

sin(3x)/sin(x) - cos(3x)/cos(x)

To simplify, we can use the trigonometric identity:

sin(2θ) = 2sin(θ)cos(θ)

Using this identity, we can rewrite the expression as:

(sin(2x)cos(x) - cos(2x)sin(x)) / sin(x)cos(x)

Now, let's further simplify by using the trigonometric identities:

sin(A - B) = sin(A)cos(B) - cos(A)sin(B)
cos(A - B) = cos(A)cos(B) + sin(A)sin(B)

Using these identities, we can rewrite the expression as:

sin(2x - x) / sin(x)cos(x)
sin(x) / sin(x)cos(x)
1 / cos(x)

Now, the conjecture we can make from this simplified expression is that the given expression is equal to the reciprocal of the cosine of x (1/cos(x)).

To prove this conjecture, we need to show that the given expression simplifies to 1/cos(x).

By using trigonometric identities and simplifying the expression, we have arrived at 1/cos(x). This confirms that the conjecture is true.

To summarize, the conjecture based on the given expression is that sin(3x)/sin(x) - cos(3x)/cos(x) simplifies to 1/cos(x), and we have proven this by applying trigonometric identities and simplifying the expression.