If a cos 3-3 a cos-sin 2- m anda sin 3-3 a cos 2-sin- n- Then m - n 2-3- m - n 2-3is equal toA- 2 a 2B- 2 a 1-3C- 2 a 2-3D- 2 a 3

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Conditional Identities

If a cos 3α+3...

Question

If $a {cos}^{3} α + 3 a cos α {sin}^{2} α = m$ and

$a {sin}^{3} α + 3 a {cos}^{2} α sin α = n$ , Then $(m + n)^{\frac{2}{3}} + (m - n)^{\frac{2}{3}}$

is equal to

$2 a^{2}$

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$2 a^{\frac{1}{3}}$

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$2 a^{\frac{2}{3}}$

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$2 a^{3}$

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Solution

The correct option is C
$2 a^{\frac{2}{3}}$

Adding and subtracting the given relation,

we get $(m + n) = a {cos}^{3} α + 3 a cos α {sin}^{2} α$

+ $3 a {cos}^{2} α . sin α + a {sin}^{3} α$

= $a (cos α + sin α)^{3}$

and similarly $(m - n) = a (cos α - sin α)^{3}$

Thus, $(m + n)^{\frac{2}{3}} + (m - n)^{\frac{2}{3}}$
$= a^{\frac{2}{3}}$ { $(cos α + sin α)^{2} + (cos α - sin α)^{2}$ }
$= a^{\frac{2}{3}}$ $[2 ({cos}^{2} α + {sin}^{2} α)]$ = $2 a^{\frac{2}{3}}$ .

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If acos3α+3acosαsin2α=m and asin3α+3acos2αsinα=n, Then (m+n)23+(m−n)23 is equal to

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