Question

1) Use double-angle identities to write the following expression, using trigonometric functions of x instead of 4x.
cos 4x


2) Use half-angle identities to write the following expression, using trigonometric functions of x instead of x/4.

sin x/4

Answer 1

cos(4x) = 2cos^2(2x)-1
cos2x = 2cos^2(x) - 1

cos(4x) = 2(2cos^2(x)-1)^2 - 1
= 2(4cos^4(x) - 4cos^2(x)+1) - 1
= 8cos^4(x) - 8cos^2(x) + 1
or,
sin^4(x) + cos^4(x) -6 sin^2(x)cos^2(x)
or a bunch of other expressions

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