If acos3-3 acosasin 2- m andasin3-3 acos2-sin- n- Then m - n 2-3- m - n 2-3is equal toA- 2 a2B- 2 a1 - 3C- 2 a2 - 3D- 2 a3

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Conditional Identities

If acos3α+3 a...

Question

If $a c o s^{3} α + 3 a c o s α s i n^{2} α$ = m and

$a s i n^{3} α + 3 a c o s^{2} α s i n α$ = n, Then $(m + n)^{\frac{2}{3}} + (m - n)^{\frac{2}{3}}$

is equal to

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Solution

The correct option is C

Adding and subtracting the given relation,

we get (m+ n) = $a c o s^{3} α + 3 a c o s α s i n^{2} α$

+ $3 a c o s^{2} α . s i n α + a s i n^{3} α$

= $a (c o s α + s i n α)^{3}$

and similarly (m - n) = $a (c o s α - s i n α)^{3}$

Thus, $(m + n)^{\frac{2}{3}} + (m - n)^{\frac{2}{3}}$

= $a^{\frac{2}{3}}$ ${(c o s α + s i n α)^{2}$ + $(c o s α - s i n α)^{2}}$

= $a^{\frac{2}{3}}$ ${2 (c o s^{2} α + s i n^{2} α)}$ = $2 a^{\frac{2}{3}}$ .

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

The correct option is C Adding and subtracting the given relation, we get (m+ n) = acos3α+3acosαsin2α + 3acos2α.sinα+asin3α = a(cosα+sinα)3 and similarly (m - n) = a(cosα−sinα)3 Thus, (m+n)23+(m−n)23 = a23 {(cosα+sinα)2 + (cosα−sinα)2} = a23 {2(cos2α+sin2α)} = 2a23.

If $a c o s^{3} α + 3 a c o s α s i n^{2} α$ = m and

$a s i n^{3} α + 3 a c o s^{2} α s i n α$ = n, Then $(m + n)^{\frac{2}{3}} + (m - n)^{\frac{2}{3}}$

is equal to