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Is the following relation true?

$\sin 3 A \cdot \sin^{3} A + \cos 3 A \cdot \cos^{3} A = \cos^{3} 2 A$

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True

False

answer is A.

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Detailed Solution

Given relation is,

\sin 3 A \cdot \sin^{3} A + \cos 3 A \cdot \cos^{3} A = \cos^{3} 2 A

Take left hand side,

\sin 3 A \cdot \sin^{3} A + \cos 3 A \cdot \cos^{3} A

We know that,

\cos 3 A = 4 \cos^{3} A - 3 \cos A

………..(1)

\sin 3 A = 3 \sin A - 4 \sin^{3} A

…………(2)
Put these values in left hand side of the given equation,

\sin 3 A \cdot \sin^{3} A + \cos 3 A \cdot \cos^{3} A

= \cos^{3} A (4 \cos^{3} A - 3 \cos A) + \sin^{3} A (3 \sin A - 4 \sin^{3} A)

= 4 \cos^{6} A - 3 \cos^{4} A + 3 \sin^{4} A - 4 \sin^{6} A

= 4 (\cos^{6} A - \sin^{6} A) - 3 (\cos^{4} A - \sin^{4} A)

= 4 \{{(\cos^{2} A)}^{3} - {(\sin^{2} A)}^{3}\} - 3 \{{(\cos^{2} A)}^{2} - {(\sin^{2} A)}^{2}\}

= 4 \{\cos^{2} A - \sin^{2} A\} \{{(\cos^{2} A)}^{2} + \cos^{2} {Asin}^{2} A + {(\sin^{2} A)}^{2}\} -

3 (\cos^{2} A - \sin^{2} A) (\cos^{2} A + \sin^{2} A)

= (\cos^{2} A - \sin^{2} A) [4 \{{(\cos^{2} A)}^{2} + \cos^{2} {Asin}^{2} A + {(\sin^{2} A)}^{2}\} - 3.1]

= (\cos^{2} A - \sin^{2} A) [4 \cos^{2} A + 4 \cos^{2} {Asin}^{2} A + 4 \sin^{4} A - 3 {(\cos^{2} A + \sin^{2} A)}^{2}]

= (\cos^{2} A - \sin^{2} A) [4 \cos^{4} A + 4 \cos^{2} {Asin}^{2} A + 4 \sin^{4} A - 3 (\cos^{4} A + 2 \cos^{2} {Asin}^{2} A + \sin^{4} A

= (\cos^{2} A - \sin^{2} A) [\cos^{4} A - 2 \cos^{2} {Asin}^{2} A + \sin^{4} A]

= (\cos^{2} A - \sin^{2} A) {(\cos^{2} A - \sin^{2} A)}^{2},

= {(\cos^{2} A - \sin^{2} A)}^{3}

= {(\cos^{2} A)}^{3}

= \cos^{3} 2 A

Thus,

\sin 3 A \cdot \sin^{3} A + \cos 3 A \cdot \cos^{3} A = \cos^{3} 2 A

So, given relation is true.
Option 1 is correct.

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