Using Complex exponential definitions of sine and cosine, prove cos &#x2061;<!-- ⁡ --> 2

Jamiya Weber

Jamiya Weber

Answered question

2022-06-25

Using Complex exponential definitions of sine and cosine, prove cos 2 θ = cos 2 θ sin 2 θ

Answer & Explanation

crociandomh

crociandomh

Beginner2022-06-26Added 19 answers

Hint:
cos 2 θ + i sin 2 θ = e 2 i θ = ( e i θ ) 2 = ( cos θ + i sin θ ) 2
Then expand the right side and compare real parts
Eden Solomon

Eden Solomon

Beginner2022-06-27Added 7 answers

cos ( 2 θ ) + i sin ( 2 θ ) = e 2 i θ = ( e i θ ) 2 = ( cos θ + i sin θ ) 2 = ( cos θ ) 2 + 2 i cos θ sin θ + i 2 ( sin θ ) 2 = ( cos θ ) 2 + 2 i cos θ sin θ ( sin θ ) 2
Equating real and imaginary parts gives us:
cos ( 2 θ ) = ( cos θ ) 2 ( sin θ ) 2 sin ( 2 θ ) = 2 cos θ sin θ

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