If a cos^3 alpha + 3a cos alpha sin^2 alpha = | vedclass

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(C) Given:$m = a \cos^3 \alpha + 3a \cos \alpha \sin^2 \alpha$$n = a \sin^3 \alpha + 3a \cos^2 \alpha \sin \alpha$Adding $m$ and $n$:$m + n = a(\cos^3

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$2a^{2/3}$

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(C) Given:$m = a \cos^3 \alpha + 3a \cos \alpha \sin^2 \alpha$$n = a \sin^3 \alpha + 3a \cos^2 \alpha \sin \alpha$Adding $m$ and $n$:$m + n = a(\cos^3 \alpha + 3 \cos \alpha \sin^2 \alpha + \sin^3 \alpha + 3 \cos^2 \alpha \sin \alpha)$$m + n = a(\cos \alpha + \sin \alpha)^3$$(m + n)^{2/3} = a^{2/3}(\cos \alpha + \sin \alpha)^2$Subtracting $n$ from $m$:$m - n = a(\cos^3 \alpha + 3 \cos \alpha \sin^2 \alpha - \sin^3 \alpha - 3 \cos^2 \alpha \sin \alpha)$$m - n = a(\cos \alpha - \sin \alpha)^3$$(m - n)^{2/3} = a^{2/3}(\cos \alpha - \sin \alpha)^2$Now,adding the two results:$(m + n)^{2/3} + (m - n)^{2/3} = a^{2/3}[(\cos \alpha + \sin \alpha)^2 + (\cos \alpha - \sin \alpha)^2]$$= a^{2/3}[(\cos^2 \alpha + \sin^2 \alpha + 2 \sin \alpha \cos \alpha) + (\cos^2 \alpha + \sin^2 \alpha - 2 \sin \alpha \cos \alpha)]$$= a^{2/3}[1 + 1] = 2a^{2/3}$

If $a \cos^3 \alpha + 3a \cos \alpha \sin^2 \alpha = m$ and $a \sin^3 \alpha + 3a \cos^2 \alpha \sin \alpha = n$,then $(m + n)^{2/3} + (m - n)^{2/3}$ is equal to:

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