How do I simplify 2 cos &#x2061;<!-- ⁡ --> ( t ) cos &#x2061;<!-- ⁡ -->

Finley Mckinney

Finley Mckinney

Answered question

2022-06-16

How do I simplify 2 cos ( t ) cos ( 2 t ) sin ( t ) sin ( 2 t )? I know this should be possible, but I don't know how.
I have tried the cos ( t ) cos ( u ) sin ( t ) sin ( u ) = cos ( t + u ), but I don't know what to do with the 2 in front of cos(t)

Answer & Explanation

assumintdz

assumintdz

Beginner2022-06-17Added 22 answers

2 cos ( t ) cos ( 2 t ) sin ( t ) sin ( 2 t ) = cos ( t ) + 3 cos ( 3 t ) 2
Proof:
2 cos ( t ) cos ( 2 t ) sin ( t ) sin ( 2 t ) = cos ( t ) + 3 cos ( 3 t ) 2
2 ( 2 cos ( t ) cos ( 2 t ) sin ( t ) sin ( 2 t ) ) = cos ( t ) + 3 cos ( 3 t )
2 ( cos ( t ) + cos ( 3 t ) sin ( t ) sin ( 2 t ) ) = cos ( t ) + 3 cos ( 3 t )
2 ( cos ( t ) + cos ( 3 t ) sin ( t ) sin ( 2 t ) ) = cos ( t ) + 3 cos ( 3 t )
2 ( cos ( t ) + cos ( 3 t ) + cos ( 3 t ) cos ( t ) 2 ) = cos ( t ) + 3 cos ( 3 t )
2 ( cos ( t ) + cos ( 3 t ) + cos ( 3 t ) cos ( t ) 2 ) = cos ( t ) + 3 cos ( 3 t )
2 ( cos ( t ) + cos ( 3 t ) + cos ( 3 t ) 2 cos ( t ) 2 ) = cos ( t ) + 3 cos ( 3 t )
2 ( cos ( t ) 2 + 3 2 cos ( 3 t ) ) = cos ( t ) + 3 cos ( 3 t )
cos ( t ) + 3 cos ( 3 t ) = cos ( t ) + 3 cos ( 3 t )
Cory Patrick

Cory Patrick

Beginner2022-06-18Added 6 answers

Just have a bit of patience:
2 cos t cos 2 t sin t sin 2 t = 2 cos t ( 2 cos 2 t 1 ) 2 sin 2 t cos t = 2 cos t ( 2 cos 2 t 1 ) 2 cos t ( 1 cos 2 t ) = 2 cos t ( 2 cos 2 t 1 1 + cos 2 t ) = 2 cos t ( 3 cos 2 t 2 )
If you had a plus, instead of minus, it would be
2 cos t cos 2 t + sin t sin 2 t = 2 cos 3 t

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