Prove the following identity using basic pythagorean sum and

Prove the following identity using basic, pythagorean, sum and difference or double/half angle identites. Show your work.

1-8sin²xcos²x=cos4x

Solution

Remember that cos(2x) = cos²(x) - sin²(x) = 2 cos²(x) - 1 = 1 - 2 sin²(x)
Sin2x=2sinxcosx

lets start with right hand side
cos(4x) = cos(2*2x)
1 - 2sin²(2x)
1 - 2[sin(2x)]²
1 - 2[2sin(x)cos(x)]²
1 - 2*4sin²(x)cos²(x)
1 - 8sin²(x)cos²(x) = left hand side
Hence proved