Question

Use the squared identities to simplify 2sin^2xsin^2x​

Answer 1

Answer:

option A  

Step-by-step explanation:

cos(4x) = cos(2x+2x) = cos(2x)*cos(2x) - sin(2x)*sin(2x)

cos(4x) = cos^2(2x) - sin^2(2x)

cos(4x) = cos^2(2x) - cos^2(2x) - 1     (using cos^2(2x) + sin^2(2x) = 1)

cos(4x) = 2cos^2(2x) - 1   (eq. 1)

cos(2x) = cos(x+x) = cos(x)*cos(x) - sin(x)*sin(x)

cos(2x) = cos^2(x) - sin^2(x)

cos(2x) = 1 - sin^2(x) - sin^2(x)     (using cos^2(x) + sin^2(x) = 1)

cos(2x) =  1 - 2sin^2(x)    (eq. 2)

Replacing eq. 2 into eq. 1:

cos(4x) = 2[1 - 2sin^2(x)]^2 - 1

cos(4x) = 2[1 - 4sin^2(x) + 4sin^4(x)] - 1

cos(4x) = 1 - 8sin^2(x) + 8sin^4(x)

cos(4x) - 1 + 8sin^2(x) = 8sin^4(x)

cos(4x)/4 - 1/4 + 2sin^2(x) = 2sin^4(x)

2sin^2(x)sin^2(x)​ = 2sin^4(x) = cos(4x)/4 - 1/4 + 2sin^2(x)

Using eq. 2:

2sin^2(x)sin^2(x)​ = cos(4x)/4 - 1/4 + 1 - cos(2x)

2sin^2(x)sin^2(x)​ = cos(4x)/4 + 3/4 - 4cos(2x)/4

2sin^2(x)sin^2(x)​ = [3 + cos(4x) - 4cos(2x)]/4