prove that cos^4 x=1/8(3+4cos2x+cos 4x)
well,
cos2x = 2cos^2x-1
cos4x = 2cos^2(2x)-1
= 2(2cos^2x-1)^2-1
now expand all that out separate out the cos^4x
To prove that cos^4(x) = 1/8(3 + 4cos(2x) + cos(4x)), we will use the double-angle and power-reduction identities for cosine.
Step 1: Start with the left side of the equation: cos^4(x)
cos^4(x) = (cos^2(x))^2
Step 2: Apply the power-reduction identity: cos^2(x) = 1/2(1 + cos(2x))
cos^4(x) = (1/2(1 + cos(2x)))^2
= 1/4(1 + 2cos(2x) + cos^2(2x))
Step 3: Rewrite cos^2(2x) using the power-reduction identity: cos^2(2x) = 1/2(1 + cos(4x))
cos^4(x) = 1/4(1 + 2cos(2x) + 1/2(1 + cos(4x)))
= 1/4(1 + 2cos(2x) + 1/2 + 1/2cos(4x))
= 1/4(3/2 + 2cos(2x) + 1/2cos(4x))
= 1/8(3 + 4cos(2x) + cos(4x))
Therefore, we have proven that cos^4(x) = 1/8(3 + 4cos(2x) + cos(4x)).