Prove sin 2 ⁡ θ + cos 4 ⁡ θ = cos 2 ⁡ θ + sin 4 ⁡ θ

Question

Prove       sin    2    ⁡  θ  +      cos    4    ⁡  θ  =      cos    2    ⁡  θ  +      sin    4    ⁡  θ

Shawn Castaneda · Accepted Answer

As       cos    2    ⁡  θ  +      sin    2    ⁡  θ  =  1 we have      cos    4    ⁡  θ  −      sin    4    ⁡  θ  =  (      cos    2    ⁡  θ  −      sin    2    ⁡  θ  )      (          cos      2        ⁡    θ    +          sin      2        ⁡    θ    )    =      cos    2    ⁡  θ  −      sin    2    ⁡  θThat is,      sin    2    ⁡  θ  +      cos    4    ⁡  θ  =      cos    2    ⁡  θ  +      sin    4    ⁡  θ

Araceli Clay · Accepted Answer

I took the long-haul approach for you since it's nice and clear to see. There is a lot of play around with the fact:

\sin^{2} θ + \cos^{2} θ = 1

rearranged into

\sin^{2} θ = 1 - \cos^{2} θ

and

\cos^{2} θ = 1 - \sin^{2} θ

We can see that:

\cos^{4} θ = \cos^{2} θ \cos^{2} θ = (1 - \sin^{2} θ) (1 - \sin^{2} θ) = 1 - 2 \sin^{2} θ + \sin^{4} θ

\sin^{2} θ + \cos^{4} θ = \cos^{2} θ + \sin^{4} θ

\sin^{2} θ + (1 - 2 \sin^{2} θ + \sin^{4} θ) = \cos^{2} θ + \sin^{4} θ

\sin^{2} θ + 1 - 2 \sin^{2} θ + \sin^{4} θ = \cos^{2} θ + \sin^{4} θ

\sin^{4} θ - \sin^{2} θ + 1 = \cos^{2} θ + \sin^{4} θ

\sin^{4} θ - (1 - \cos^{2} θ) + 1 = \cos^{2} θ + \sin^{4} θ

\sin^{4} θ - 1 + \cos^{2} θ + 1 = \cos^{2} θ + \sin^{4} θ

\sin^{4} θ + \cos^{2} θ = \cos^{2} θ + \sin^{4} θ

Prove sin 2 </msup> ⁡ θ + cos

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