how do you simplify, sin^2 x - 9/2cos x + 1 * 10cos x + 5/3sin x + 9? please help!?

4+2=6...there, that's a lot simpler.

First, do you recognize that sin² x - 9 is the difference of two squares? It factors into (sin x + 3)(sin x - 3). Also, we can factor the numerator and the denominator of the second fraction:

10cos x + 5 = 5(2cos x + 1)

3sin x + 9 = 3(sin x + 3).

After doing all the above, we can rewrite the original product like this:

[(sin x + 3)(sin x - 3) / (2cos x + 1)] • [5(2cos x + 1) / 3(sin x + 3)].

Notice now that (sin x + 3) in the numerator of the first fraction cancels its counterpart in the denominator of the second fraction. Also, we see that (2cos x + 1) in the denominator of the first fraction cancels its counterpart in the numerator of the second fraction. So, we are left with this:

(sin x - 3) • (5/3) = 5/3 (sin x - 3).

We can check the result by letting x be a convenient value. If the result above is correct, then when we plug x = 0° and x = 90° into the original expression and into the resulting simplified expression, the two values returned should be equal.

Let x = 0°, and x = 90°.

x = 0°, using original expression:

(sin² x - 9) / (2cos x + 1) • (10cos x + 5) / (3sin x + 9) =

[(0 - 9) / (2 + 1)] • [10(1) + 5 / 3(0) + 9] =

(-9 / 3) • (15 / 9) =

-3 • (15 / 9) =

-45 / 9 =

-5.

Now we use the resulting expression:

5/3 (sin x - 3) =

5/3 (0 - 3) =

(5/3) • (-3) =

5 • (-3/3) =

5 • (-1) =

-5.

Now let x = 90°, using the original expression:

(sin² x - 9) / (2cos x + 1) • (10cos x + 5) / (3sin x + 9) =

[(1 - 9) / (0 + 1)] • [(0 + 5) / (3 + 9)] =

(-8 / 1) • (5 / 12) =

-40 / 12 =

-10/3.

Now we use the simplified expression:

5/3 (sin x - 3) =

5/3 • (1 - 3) =

5/3 • (-2) =

-10/3.

Both values for x led to the same result in each expression, so the simplified answer is correct.