13-4(1-2sin²x)= 12sinx
13-4+8sin²x=12sinx
8sin²x-12sinx+9=0

sinx=-b+-√b²-4ac/2a
= +12+-√-144/2×8
= 12+-12i/16
= 3+-3i/4

**Solution:**

To solve the equation 13 - 4cos^2(x) = 12sin(x), we will follow the steps mentioned below:

**Step 1: Simplify the equation**

Let's simplify the equation by using the identity cos^2(x) = 1 - sin^2(x):

13 - 4(1 - sin^2(x)) = 12sin(x)

Simplifying further:

13 - 4 + 4sin^2(x) = 12sin(x)

9 + 4sin^2(x) - 12sin(x) = 0

**Step 2: Substitute sin(x) with a new variable**

Let's substitute sin(x) with a new variable, say t. So, the equation becomes:

4t^2 - 12t + 9 = 0

**Step 3: Solve the quadratic equation**

We can solve this quadratic equation by factoring or using the quadratic formula. Let's use the quadratic formula:

t = (-b ± sqrt(b^2 - 4ac))/2a

Here, a = 4, b = -12, and c = 9. Substituting these values into the quadratic formula:

t = (-(-12) ± sqrt((-12)^2 - 4(4)(9)))/(2(4))

Simplifying further:

t = (12 ± sqrt(144 - 144))/8

t = (12 ± sqrt(0))/8

**Step 4: Find the values of t**

Since the discriminant (144 - 144) is zero, the equation has only one real root:

t = 12/8

t = 3/2

**Step 5: Substitute back the value of t to find sin(x)**

Now, we substitute the value of t back into the equation sin(x) = t:

sin(x) = 3/2

However, sin(x) cannot be greater than 1, so there is no solution for sin(x) = 3/2.

Therefore, the given equation 13 - 4cos^2(x) = 12sin(x) has no solution.

So, the solution to the equation is the empty set, {} or Ø.