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If $m = a cos 3 θ + 3 a cos θ sin 2 θ$ and $n = a sin 3 θ$ $+ 3 a cos 2 θ sin θ,$ then what will be the value of $(m + n) 2 / 3 + (m − n) 2 / 3$ ?

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2 a 2 / 3

3 a 2 / 3

2 a 3 / 2

3 a 3 / 2

answer is A.

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Detailed Solution

Given that,

m = a cos 3 θ + 3 a cos θ sin 2 θ

n = a sin 3 θ + 3 a cos 2 θ sin θ

Now,

m + n = a cos 3 θ + sin 3 θ + 3 cos θ sin θ cos θ + sin θ

= a cos θ + sin θ cos 2 θ + sin 2 θ − cos θ sin θ + 3 cos θ sin θ

= a cos θ + sin θ cos 2 θ + sin 2 θ + 2 cos θ sin θ

= a cos θ + sin θ cos θ + sin θ 2

= a cos θ + sin θ 3

Using

a 3 − b 3 = a − b a 2 + b 2 + a b; a 2 + b 2 − 2 a b = (a − b) 2

m − n = a cos 3 θ − sin 3 θ + 3 cos θ sin θ sin θ − cos θ

= a cos θ − sin θ cos 2 θ + sin 2 θ + cos θ sin θ − 3 cos θ sin θ cos θ − sin θ

= a cos θ − sin θ cos 2 θ + sin 2 θ + cos θ sin θ − 3 cos θ sin θ

= a cos θ − sin θ cos 2 θ + sin 2 θ − 2 cos θ sin θ

= a cos θ − sin θ cos θ − sin θ 2

= a cos θ − sin θ 3

Substituting the values in

m + n 2 / 3 + m − n 2 / 3

= a cos θ + sin θ 3 2 / 3 + a cos θ − sin θ 3 2 / 3

= a 2 / 3 cos θ + sin θ 2 + cos θ − sin θ 2

= a 2 / 3 2 cos 2 θ + 2 sin 2 θ

2 a 2 / 3

Thus the value is

2 a 2 / 3

.
Therefore the correct option is 1.

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