1) To solve the equation 2cos^2x = 1 + sin x, we can use the Pythagorean identity cos^2x = 1 - sin^2x.
Substitute cos^2x = 1 - sin^2x into the equation:
2(1 - sin^2x) = 1 + sin x
Simplify:
2 - 2sin^2x = 1 + sin x
Rearrange and rewrite in quadratic form:
2sin^2x + sin x - 1 = 0
Now, this is a quadratic equation in terms of sin x. Solve for sin x using the quadratic formula or by factoring.

2) To solve the equation 5^x + 5^(x+1) = 30, we first simplify the equation.
Rearrange the equation as 5^x + 55^x = 30
Combine like terms:
65^x = 30
Divide both sides by 6:
5^x = 5
Now, we can see that x = 1 satisfies the equation. So, the solution to the equation is x = 1.