Question: Prove sin 3 </msup> &#x2061;<!-- ⁡ -->

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Answered question

2022-06-10

Question: Prove
sin 3 ( x ) cos 3 ( x ) sin ( x ) + cos ( x ) = csc 2 ( x ) cot ( x ) 2 cos 2 ( x ) 1 cot 2 ( x )

Answer & Explanation

Aiden Norman

Aiden Norman

Beginner2022-06-11Added 16 answers

You are correct except one sign :
1 cos ( x ) sin ( x ) 2 cos 2 ( x ) sin 2 ( x ) ( sin ( x ) cos ( x ) ) ( sin ( x ) + cos ( x ) )
Now using
1 = cos 2 ( x ) + sin 2 ( x )
1 cos ( x ) sin ( x ) 2 cos 2 ( x ) sin 2 ( x ) = ( 1 2 cos ( x ) sin ( x ) ) ( 1 + cos ( x ) sin ( x ) ) = ( cos 2 ( x ) + sin 2 ( x ) 2 sin ( x ) cos ( x ) ) ( cos 2 ( x ) + sin 2 ( x ) + cos ( x ) sin ( x ) ) = ( sin ( x ) cos ( x ) ) ( sin ( x ) cos ( x ) ) ( sin 2 ( x ) + sin ( x ) cos ( x ) + cos 2 ( x ) ) = ( sin ( x ) cos ( x ) ) ( sin 3 ( x ) cos 3 ( x ) )

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