y=2cot2x

and?

To simplify the expression y = 2cot(2x), we need to understand the concept of cotangent.

The cotangent function (cot) is defined as the reciprocal of the tangent function (tan). It represents the ratio of the adjacent side to the opposite side in a right triangle.

In this case, we have y = 2cot(2x), where 2x is in the angle measure. To simplify further, we can use the identity:

cot(2x) = cot^2(x) - 1

Substituting this identity into our expression, we have:

y = 2(cot^2(x) - 1)

Now, let's break it down step by step to simplify it even further.

Step 1: Simplify cot^2(x)
- We know that cot^2(x) = 1/(tan^2(x)).
- Recall that the tangent function (tan) is the ratio of the opposite side to the adjacent side in a right triangle.
- Therefore, tan^2(x) = (opposite side)^2 / (adjacent side)^2.
- Similarly, cot^2(x) = (adjacent side)^2 / (opposite side)^2.
- Hence, cot^2(x) = 1 / tan^2(x).

Therefore, substituting this back into our expression, we have:

y = 2(1 / tan^2(x)) - 1

Step 2: Simplify the expression further
- We know that 1 / tan^2(x) can be simplified as sec^2(x) − 1.
- The secant function (sec) is defined as the reciprocal of the cosine function (cos).
- Hence, sec^2(x) − 1 = 1 / cos^2(x) − 1.

Substituting this back into our expression, we get:

y = 2(1 / cos^2(x)) - 1

To summarize, we have simplified the expression y = 2cot(2x) to:

y = 2(1 / cos^2(x)) - 1