Prove that (3cos^2x + 8sinx-8/cos^2x) = (3sinx-5/sinx+1)

your lack of the proper use of brackets make your equation much too ambiguous.

Furthermore, the brackets you do use, make no difference to the meaning.
If they are removed, there is no change in the order of operation, but ....

on the left side, who is divided by cos^2 x ?
is it -8/cos^2 x ? or
(8sinx-8)/cos^2 x or
(3cos^2x + 8sinx-8)/cos^2 x

each would give you a different result, the same is just as confusing on the right side.

(3cos^2x + 8sinx-8)/cos^2x = (3sinx-5)/sinx+1

LS = (3(1-sin^x) + 8x - 8)/cos^2x

= (3-3sin^x+8x-8)/cos^2x
= -(3sin^2x - 8x + 5)/((1-sinx)(1+sinx))
= -(3sinx-5)(sinx-1)/((1-sinx)(1+sinx))
= (3sinx-5)(1 - sinx)/((1-sinx)(1+sinx))
= (3sinx-5)/(1+sinx)
= RS

To prove that (3cos^2x + 8sinx-8/cos^2x) = (3sinx-5/sinx+1), we need to simplify both sides of the equation and show that they are equal.

Let's start by simplifying the left side of the equation:

(3cos^2x + 8sinx - 8) / cos^2x

Using the identity cos^2x = 1 - sin^2x, we can substitute it into the equation:

(3(1-sin^2x) + 8sinx - 8) / (1-sin^2x)

Expanding the equation:

(3 - 3sin^2x + 8sinx - 8) / (1 - sin^2x)

Rearranging the terms:

(-3sin^2x + 8sinx - 5) / (1 - sin^2x)

Using the identity 1 - sin^2x = cos^2x, we can substitute it into the equation:

(-3sin^2x + 8sinx - 5) / cos^2x

Now, let's simplify the right side of the equation:

(3sinx - 5) / (sinx + 1)

To make them comparable, we can multiply the numerator and denominator of the right side by cos^2x:

[(3sinx - 5) * cos^2x] / [(sinx + 1) * cos^2x]

Expanding the equation:

[(3sinx * cos^2x - 5 * cos^2x)] / [(sinx * cos^2x + cos^2x)]

Using the identity sinx * cos^2x = sinx - sin^3x, we can substitute it into the equation:

[(3sinx - 5 * cos^2x)] / [sinx - sin^3x + cos^2x]

Rearranging the terms:

[3sinx - 5 * cos^2x] / [sinx + cos^2x - sin^3x]

Using the identity sinx + cos^2x = 1 + sin^2x, we can substitute it into the equation:

[3sinx - 5 * cos^2x] / [1 + sin^2x - sin^3x]

Now, we can see that the left side of the equation and the right side of the equation have the same expression:

(-3sin^2x + 8sinx - 5) / cos^2x = [3sinx - 5 * cos^2x] / [1 + sin^2x - sin^3x]

We have successfully proven that (3cos^2x + 8sinx-8/cos^2x) = (3sinx-5/sinx+1).