Question

How do you verify the identity `(cos4x+cos2x)/(sin4x+sin2x)=cot3x`?

Answer 1

Please see below.

Explanation:

We use the identities `cosA+cosB=2cos((A+B)/2)cos((A-B)/2)`

and `sinA+sinB=2sin((A+B)/2)cos((A-B)/2)`

Hence `cos4x+cos2x=2cos((4x+2x)/2)cos((4x-2x)/2)`

= `2cos3xcosx`

and `sin4x+sin2x=2sin((4x+2x)/2)cos((4x-2x)/2)`

= `2sin3xcosx`

Hence `(cos4x+cos2x)/(sin4x+sin2x)=(2cos3xcosx)/(2sin3xcosx)=cot3x`