Rewrite as single trig function: sin(8x)cosx-cos(8x)sinx
I know I can simplify sin(8x) into 4sin2xcos2xcos4x, but I'm stuck after that
how about using
sin(a-b) = sina cosb - cosa sinb
To rewrite the expression sin(8x)cos(x) - cos(8x)sin(x) as a single trigonometric function, we can apply the product-to-sum identities for sine and cosine.
First, using the product-to-sum identity for sine, we have:
sin(A)cos(B) - cos(A)sin(B) = sin(A - B).
Applying this identity to sin(8x)cos(x) - cos(8x)sin(x), we have:
sin(8x)x - cos(8x)x = sin(8x - x) = sin(7x).
Therefore, the expression sin(8x)cos(x) - cos(8x)sin(x) can be rewritten as sin(7x).
To simplify the expression sin(8x)cos(x) - cos(8x)sin(x), you can use the identities sin(A + B) = sinAcosB + cosAsinB and cos(A + B) = cosAcosB - sinAsinB.
First, let's rewrite sin(8x) as 4sin(2x)cos(2x)cos(4x) by using the double-angle formula for sine: sin(2A) = 2sin(A)cos(A). So, sin(8x) = 2sin(4x)cos(4x), and by applying the double-angle formula again to sin(4x), we have sin(4x) = 2sin(2x)cos(2x). Substituting this back, we get sin(8x) = 2(2sin(2x)cos(2x))cos(4x), which simplifies to 4sin(2x)cos(2x)cos(4x).
Next, let's rewrite cos(8x) as cos(4x + 4x) using the double-angle formula for cosine: cos(2A) = cos^2(A) - sin^2(A). So, cos(8x) = cos^2(4x) - sin^2(4x), and by applying the double-angle formula again to cos(4x), we have cos(4x) = cos^2(2x) - sin^2(2x). Substituting this back, we get cos(8x) = (cos^2(2x) - sin^2(2x))^2 - sin^2(4x), which can be simplified further.
Now, using the identities sin^2(A) = 1 - cos^2(A) and cos^2(A) = 1 - sin^2(A), we can simplify cos(8x) as follows:
cos(8x) = ((cos^2(2x) - sin^2(2x))^2 - (1 - cos^2(2x) + sin^2(2x))^2)
= ((1 - sin^2(2x) - sin^2(2x))^2 - (1 - (1 - sin^2(2x) + sin^2(2x)))^2)
= ((1 - 2sin^2(2x))^2 - (1 - 2sin^2(2x))^2)
= 0.
Finally, substituting 0 for cos(8x), we have sin(8x)cos(x) - cos(8x)sin(x) = 4sin(2x)cos(2x)cos(4x) - 0 = 4sin(2x)cos(2x)cos(4x).
So, the simplified expression is 4sin(2x)cos(2x)cos(4x).
using sin(a-b) =sina cosb - cosa sinb
sin (8x-1x) = sin8x cosx - cos8x sinx
therefore the answer is sin7x