1-3cos=sin squared

You need to define the variables, such as x, y etc.
If you are trying to solve
1 - 3 cos x = sin^2 x for whatever x is, then you are not talking about an inverse function.

To solve the equation 1 - 3cos(x) = sin^2(x), you need to find the values of x that satisfy the equation. Since the equation involves both trigonometric functions, there are a few approaches you can take to find the solutions.

One way is to use fundamental trigonometric identities to rewrite the equation in terms of a single trigonometric function. For example, the Pythagorean identity sin^2(x) + cos^2(x) = 1 can be used to replace sin^2(x) in the equation. By rearranging the equation, we can get:

1 - 3cos(x) = 1 - cos^2(x)

Next, we can move all terms to one side of the equation:

cos^2(x) - 3cos(x) + 1 - 1 = 0

Simplifying further, we get:

cos^2(x) - 3cos(x) = 0

Now, we have a quadratic equation in terms of cos(x). To solve it, you can factor or use the quadratic formula:

cos(x)(cos(x) - 3) = 0

The solutions to this equation are cos(x) = 0 and cos(x) = 3.

However, the range of the cosine function is -1 to 1, so cos(x) = 3 has no solution. Therefore, we only have cos(x) = 0 as a valid solution. This occurs when x is a multiple of π/2.

In summary, the solutions to the equation 1 - 3cos(x) = sin^2(x) are x = π/2 + kπ, where k is an integer.