Find all exact solutions
2cos^2x=sinx+1
Please show work thank you!

Step-by-step explanation:

We are given the equation:

And we want to find all solutions for x.

First, we should put the equation into terms of only one trigonometric ratio.

Since we are given cos²(x), we can turn this into sine. Recall the Pythagorean Identity which states:

Therefore:

By substitution:

Distribute:

Isolate the equation:

We can factor:

Zero Product Property:

Solve for each case:

We can use the unit circle.

sin(x) = 1/2 for every π/6 and 5π/6. So, it will continue every 2π.

sin(x) = -1 every 3π/2. And this will also continue every 2π.

Hence, our solutions are:

Note:

If you only need the solutions within the interval [0, 2π), then it is:

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