Rewrite 3 sin^4(2x) − 2 cos^2(4x) in terms of the first power of cosine

Question

Please show step by step instructions, thank you!

Answer 1

Hi Lily,

you can reduce these terms several ways, If the question is ok with having the 4x and 2x inside the cosine then you just need the following altered form of the half angle formula

sin²(∅) = (1/2)(1-cos(2∅))

cos²(∅) = (1/2)(1+cos(2∅))

3sin⁴(2x) - 2cos²(4x)

3sin²(2x)sin²(2x) - 2cos²(4x)

3[(1/2)(1 - cos(2*(2x)))(1/2)(1 - cos(2*(2x)))] - 2*[ (1/2)(1 + cos(4*(2x))) ]

(3/4) * (1 - cos(4x))(1 - cos(4x)) - (1 + cos(8x))

(3/4) * [1*1 - 1*cos(4x) - cos(4x)*1 + (cos(4x))²] - 1 - cos(8x)

3/4 - (3/4)*2cos(4x) + (3/4)cos²(4x) - 1 - cos(8x)

-1/4 - (3/4)*2cos(4x) + (3/4)(1/2)(1 + cos(4*2x)) - cos(8x)

-1/4 - (3/2)cos(4x) + (3/8)(1 + cos(8x)) - cos(8x)

-1/4 - (3/2)cos(4x) + 3/8 + (3/8)cos(8x) - (8/8)cos(8x)

-2/8 + 3/8 - (3/2)cos(4x) + (3/8)cos(8x) - (8/8)cos(8x)

1/8 - (3/2)cos(4x) - (5/8)cos(8x)

I may have missed a 2 or something, so please check my work. Let me know if you have other questions!