The formula for sin 2x can be derived using various trigonometric identities and relationships. One way to express sin 2x is in terms of the tan function.

To understand this, let’s start with the double-angle formula for sine: sin 2x = 2sin x cos x.

Now, we can express sin x and cos x in terms of the tan function. Recall that tan x = sin x / cos x. Rearranging this equation, we get sin x = tan x cos x. Similarly, we can rewrite cos x as cos x = 1 / sec x = 1 / (1 / cos x) = cos x / 1.

Substituting these expressions into the double-angle formula for sine, we have:

Sin 2x = 2(tan x cos x)(cos x / 1) = 2tan x cos^2 x.

Now, we can simplify this further using the identity cos^2 x = 1 – sin^2 x. Substituting this into the equation, we get:

Sin 2x = 2tan x (1 – sin^2 x).

To eliminate the sin^2 x term, we can use the Pythagorean identity sin^2 x + cos^2 x = 1. Rearranging this equation, we have sin^2 x = 1 – cos^2 x. Substituting this into the equation above, we get:

Sin 2x = 2tan x (1 – (1 – cos^2 x)) = 2tan x (cos^2 x).

Simplifying further, we have:

Sin 2x = 2tan x cos^2 x.

Therefore, the formula for sin 2x in terms of the tan function is sin 2x = 2tan x cos^2 x.

It’s important to note that this is just one way to express sin 2x in terms of the tan function. There are other identities and relationships that can be used to derive alternative formulas.