Prove that cos ⁡ ( 5 A ) = 16 cos 5 ⁡ ( A ) − 20 cos 3 ⁡ ( A ) + 5 cos ⁡ ( A )

Question

Prove that   cos  ⁡  (  5  A  )  =  16      cos    5    ⁡  (  A  )  −  20      cos    3    ⁡  (  A  )  +  5  cos  ⁡  (  A  )

Leland Ochoa · Accepted Answer

A general way to approach this is to use Euler’s formula and the binomial theorem, as follows.

\cos (5 A) = \frac{e^{5 A i} + e^{- 5 A i}}{2} = \frac{(e^{A i})^{5} + (e^{- A i})^{5}}{2} = \frac{(\cos A + i \sin A)^{5} + (\cos A - i \sin A)^{5}}{2} .

Let

x = \cos A

and

y = \sin A

, and note for later that

y^{2} = 1 - x^{2}

. Now you can write

2 \cos (5 A) = (x + i y)^{5} + (x - i y)^{5} .

Expand each fifth power using the binomial theorem. No odd powers of y or imaginary terms should remain, and you can replace

y^{2}

with

1 - x^{2}

and simplify to get your result.

gledanju0 · Accepted Answer

Note that   cos  ⁡  (  a  +  b  )  +  cos  ⁡  (  a  −  b  )  =  2  cos  ⁡  a  cos  ⁡  b, so  cos  ⁡  5  x  +  cos  ⁡  3  x  =  2  cos  ⁡  x  cos  ⁡  4  x    ⟹    cos  ⁡  5  x  =  2  cos  ⁡  4  x  cos  ⁡  x  −  cos  ⁡  3  x  cos  ⁡  4  x  +  cos  ⁡  2  x  =  2  cos  ⁡  x  cos  ⁡  3  x    ⟹    cos  ⁡  4  x  =  2  cos  ⁡  3  x  cos  ⁡  x  −  cos  ⁡  2  x  cos  ⁡  4  x  +  cos  ⁡  2  x  =  2  cos  ⁡  x  cos  ⁡  3  x    ⟹    cos  ⁡  4  x  =  2  cos  ⁡  3  x  cos  ⁡  x  −  cos  ⁡  2  x  cos  ⁡  2  x  +  cos  ⁡  0  x  =  2  cos  ⁡  x  cos  ⁡  x    ⟹    cos  ⁡  2  x  =  2      cos    2    ⁡  x  −  1Gradually pushing these relations upwards allows us to express cosnx in terms of just cosx, culminating in the result you obtain.

Prove that cos ⁡ ( 5 A ) = 16 cos 5 </msu

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