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

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Answer 1 · 2017-01-02T10:13:33

Please see below.

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

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