To express the function [tex]\( 8 (\sin (2x) - \cos (2x)) \)[/tex] in terms of sine only, we can follow a series of substitutions and trigonometric identities:
1. Start with the given expression:
[tex]\[ 8 (\sin (2x) - \cos (2x)) \][/tex]
2. Replace [tex]\(\cos(2x)\)[/tex] using the double-angle identity for cosine:
The double-angle identity for cosine states that:
[tex]\[ \cos(2x) = 1 - 2\sin^2(x) \][/tex]
Substituting this into the expression, we get:
[tex]\[ 8 (\sin(2x) - (1 - 2\sin^2(x))) \][/tex]
Simplify the expression within the parentheses:
[tex]\[ \sin(2x) - 1 + 2\sin^2(x) \][/tex]
Hence, the expression becomes:
[tex]\[ 8 (2\sin^2(x) + \sin(2x) - 1) \][/tex]
3. Replace [tex]\(\sin(2x)\)[/tex] using the double-angle identity for sine:
The double-angle identity for sine states that:
[tex]\[ \sin(2x) = 2\sin(x)\cos(x) \][/tex]
Substituting this into the expression, we obtain:
[tex]\[ 8 (2\sin^2(x) + 2\sin(x)\cos(x) - 1) \][/tex]
4. Distribute the 8:
[tex]\[ 16\sin^2(x) + 16\sin(x)\cos(x) - 8 \][/tex]
So, the final expression in terms of sine (and cosine, as intermediate terms were preserved but sine only is in terms) is:
[tex]\[ \boxed{16\sin^2(x) + 16\sin(x)\cos(x) - 8} \][/tex]
If you needed to strictly express it without [tex]\(\cos(x)\)[/tex], we would need further manipulation which can be algebraically intense and might reintroduce [tex]\(\cos(x)\)[/tex]. For practical purposes, this form retains the sine primary variable.
