Prove that cos4 theta 38 12 cos2 theta 18 cos4 theta Expr

Prove that cos^4 theta = 3/8 - 1/2 cos(2 theta) + 1/8 cos(4 theta). Express the product sin(4 theta) cos (theta) as a sum containing only sines or cosine. Express the sum cos(4 theta) + cos(3 theta) as a product of sines and/or cosines.

Solution

6) cos⁴ = 3/8 – ½*cos2 + 1/8*cos4

LHS = cos⁴ = (cos² )² = ¼*[1 + cos2]^2 = ¼ *[1 + 2* cos2 + cos²2]

              = ¼ *[1 + 2* cos2 + ½ *(cos4 + 1)]

              = ¼ *[3/2 + 2* cos2 + ½ * cos4]

              = 3/8 + ½*cos2 + 1/8*cos4

7) Using 2 sin A cos B = sin(A + B) + sin(A B)

sin4*cos = ½ *[sin(4 + ) + sin(4 - )] = ½ *[ sin5 + sin3]

8) Using cos A + cos B = 2*cos(A + B)/2* cos(A – B)/2

cos4 + cos3 = 2*cos7/2 * cos/2