Prove That Cos3A = 4Cos A - 3Cos A
That is not correct.
cos 3 A = cos ( 2 A + A )
= cos ( 2 A ) ∙ cos A - sin ( 2 A ) ∙ sin A =
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Remark:
cos ( 2 A ) = cos² A - sin² A =
cos² A - ( 1 - cos² A ) =
cos² A - 1 + cos² A =
2 cos² A - 1
sin ( 2 A ) = 2 ∙ sin A ∙ cos A
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( 2 cos² A - 1 ) ∙ cos A - 2 sin A ∙ cos A ∙ sin A =
2 cos² A ∙ cos A - 1 ∙ cos A - 2 sin A ∙ sin A ∙ cos A =
2 cos³ A - cos A - 2 ∙ sin² A ∙ cos A =
2 cos³ A - cos A - 2 ∙ cos A ∙ sin² A =
2 cos³ A - cos A - 2 ∙ cos A ∙ ( 1 - cos² A ) =
2 cos³ A - cos A - 2 ∙ cos A ∙ 1 - 2 ∙ cos A ∙ ( - cos² A ) =
2 cos³A - cos A - 2 ∙ cos A + 2 ∙ cos³A =
2 cos³A + 2 ∙ cos³A - cos A - 2 ∙ cos A =
4 cos³ A - 3 cos A
So:
cos 3 A = 4 cos³ A - 3 cos A
To prove the equation Cos3A = 4Cos A - 3Cos A, we'll use the trigonometric identity for the cosine of the triple angle, which states:
Cos3A = 4Cos³A - 3CosA
First, let's simplify the right-hand side of the equation, which is 4Cos A - 3Cos A:
4Cos A - 3Cos A
Combining like terms, we have:
Cos A
Now, applying the simplified right-hand side to the original equation, we get:
Cos3A = Cos A
To further prove that Cos3A = Cos A, we can use the unit circle or the periodicity of the cosine function.
Using the unit circle, we can define the cosine of an angle A in terms of its coordinates on the unit circle. When we examine the unit circle, we find that the x-coordinate of the point corresponding to an angle A is equal to the cosine of A. Similarly, the x-coordinate of the point corresponding to an angle 3A is equal to the cosine of 3A.
Since the x-coordinate of the point corresponding to an angle 3A is the same as the x-coordinate of the point corresponding to an angle A, we conclude that:
Cos3A = Cos A
Therefore, we have proved that Cos3A = 4Cos A - 3Cos A is a true statement.