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3.Trigonometrical Ratios, Functions and Identities
hard
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
A
$2{a^2}$
B
$2{a^{1/3}}$
C
$2{a^{2/3}}$
D
$2{a^3}$
Solution
(c) Adding and subtracting the given relation, we get
$(m + n) = a{\cos ^3}\alpha + 3a\cos \alpha \,{\sin ^2}\alpha $ $ + 3a{\cos ^2}\alpha .\sin \alpha + a{\sin ^3}\alpha $
$ = a{(\cos \alpha + \sin \alpha )^3}$
and similarly $(m – n) = a\,\,{(\cos \alpha – \sin \alpha )^3}$
Thus, ${(m + n)^{2/3}} + {(m – n)^{2/3}}$
$ = {a^{2/3}}{\{ \cos \alpha + \sin \alpha )^2} + {(\cos \alpha – \sin \alpha )^2}\} $
$ = {a^{2/3}}\{ 2({\cos ^2}\alpha + {\sin ^2}\alpha )\} = 2{a^{2/3}}$.
Standard 11
Mathematics
