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

pouzdrotf

pouzdrotf

Answered

2022-06-30

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

Answer & Explanation

Shawn Castaneda

Shawn Castaneda

Expert

2022-07-01Added 17 answers

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 θ
That is,
sin 2 θ + cos 4 θ = cos 2 θ + sin 4 θ
Araceli Clay

Araceli Clay

Expert

2022-07-02Added 2 answers

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

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