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Sign of Trigonometric Ratios in Different Quadrants

If acos3α +...

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 $(m + n)^{2 / 3} + (m - n)^{2 / 3}$ is equal to

A

2 a^{2}

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B

2 a^{1 / 3}

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C

2 a^{2 / 3}

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D

2 a^{3}

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Solution

The correct option is D $2 a^{2 / 3}$
$a {c o s}^{3} α + 3 a c o s α {s i n}^{2} α = m$ ...(1)

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

Adding (1) and (2)

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

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

$= a {(c o s α + s i n α)}^{3}$

Subtracting (2) from (1)

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

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

$= a {(c o s α - s i n α)}^{3}$

Therefore,

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

$= a^{\frac{2}{3}} (1 + 2 c o s α s i n α + 1 - 2 c o s α s i n α) = 2 a^{\frac{2}{3}}$

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