We can simplify this trigonometric expression by using the double angle identity for cosine:

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

Substitute this identity into the original expression:

6cos^2(x) + 5cos(π/2 - x) = 7

6cos^2(x) + 5(sin(x)) = 7

Now, we can use the double angle identity for sine:

sin(π/2 - x) = cos(x)

Substitute this identity into the expression:

6cos^2(x) + 5cos(x) = 7

Now, we have a quadratic equation in terms of cos(x):

6cos^2(x) + 5cos(x) - 7 = 0

Now, we can solve this quadratic equation for cos(x). We can use the quadratic formula:

cos(x) = [-b ± sqrt(b^2 - 4ac)] / 2a

In this case, a = 6, b = 5, and c = -7. Plugging in these values, we get:

cos(x) = [-5 ± sqrt(5^2 - 46(-7))] / 2*6
cos(x) = [-5 ± sqrt(25 + 168)] / 12
cos(x) = [-5 ± sqrt(193)] / 12

Therefore, the solutions for cos(x) are:

cos(x) = (-5 + sqrt(193)) / 12 or cos(x) = (-5 - sqrt(193)) / 12

These are the two possible values for cos(x) that satisfy the given trigonometric equation.