8sin^4(x)
To simplify the expression 8sin^4(x), we can start by using the trigonometric identity: sin^2(x) = (1 - cos(2x))/2.
Let's substitute this identity into the expression:
8sin^4(x) = 8(sin^2(x))^2 = 8[(1 - cos(2x))/2]^2
Next, we can simplify further by squaring both the numerator and the denominator inside the square brackets:
8[(1 - cos(2x))/2]^2 = 8(1 - cos(2x))^2/4
Now, we can simplify the expression inside the parentheses by expanding the square:
8(1 - cos(2x))^2/4 = 8(1 - 2cos(2x) + cos^2(2x))/4
Finally, we can simplify the expression further by dividing both the numerator and the denominator by 4:
8(1 - 2cos(2x) + cos^2(2x))/4 = 2(1 - 2cos(2x) + cos^2(2x))
Therefore, the simplified form of 8sin^4(x) is 2(1 - 2cos(2x) + cos^2(2x)).