# Thread: Trig simplification (this one is tough)

1. ## Trig simplification (this one is tough)

f(x)= (4cos (x) -cos (3x))^2 + (3sin (x) -sin (3x))^2
f(x) has to be simplified to the form: f(x)= a cos ^2 (2x) + b cos (2x) + c where a,b,and c are real numbers.

so far I expanded the two parentheticals to: f(x)= 16 cos ^2 (x) - 8 cos(x) cos(3x) + cos ^2 (3x) + 9 sin^2 (x) -6 sin (x) sin (3x) + sin ^2 (3x)

after this I'm stumped.

I tried merging the 16cos ^2 (x) + sin ^2 (3x) to a 7 cos ^2(x) + 9, and doing the same with the sin ^2 (3x) + cos ^2 (3x) = 1. But it isn't taking me anywhere.

If someone can solve this with work, I would highly appreciate it!

Thanks
-kaplac

2. Hello kaplac
Originally Posted by kaplac
f(x)= (4cos (x) -cos (3x))^2 + (3sin (x) -sin (3x))^2
f(x) has to be simplified to the form: f(x)= a cos ^2 (2x) + b cos (2x) + c where a,b,and c are real numbers.

so far I expanded the two parentheticals to: f(x)= 16 cos ^2 (x) - 8 cos(x) cos(3x) + cos ^2 (3x) + 9 sin^2 (x) -6 sin (x) sin (3x) + sin ^2 (3x)

after this I'm stumped.

I tried merging the 16cos ^2 (x) + sin ^2 (3x) to a 7 cos ^2(x) + 9, and doing the same with the sin ^2 (3x) + cos ^2 (3x) = 1. But it isn't taking me anywhere.

If someone can solve this with work, I would highly appreciate it!

Thanks
-kaplac
Using the identities
$\displaystyle \cos3x = 4\cos^3x-3\cos x$; and

$\displaystyle \sin 3x = 3\sin x - 4\sin^3x$, we get:
$\displaystyle f(x) = (7\cos x - 4\cos^3x)^2+(4\sin^3x)^2$

$\displaystyle =16\cos^6x-56\cos^4x+49\cos^2x+16(1-\cos^2x)^3$

$\displaystyle =16\cos^6x-56\cos^4x+49\cos^2x+16(1-3\cos^2x+3\cos^4x-\cos^6x)$

$\displaystyle = -8\cos^4x+\cos^2x+16$

Now $\displaystyle a\cos^22x+b\cos2x+c = a(2\cos^2x-1)^2 +b(2\cos^2x-1) + c$
$\displaystyle =4a\cos^4x+(2b-4a)\cos^2x+a-b+c$
Equating coefficients gives
$\displaystyle a = -2, b = -\frac72, c=\frac{29}{2}$
But check my working!