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 ) 2 3 + ( m − n ) 2 3 is equal to Options: A . &nbsp 2 a 2 B . &nbsp 2 a 1 3 C . &nbsp 2 a 2 3 D . &nbsp 2 a 3

If acos3α+3acosαsin2α=m and asin3α+3acos2αsin&#

Discussion Forum : Trigonometric Functions

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

Options:

A .

2 a^{2}

B .

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

C .

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

D .

2 a^{3}

Answer: Option C
:
C

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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