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  4. If A(n)= sin n α+ cos n α, then A( 1 ) A(4)+A(2) A(5)=

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Q. If $A(n)=\sin ^{n} \alpha+\cos ^{n} \alpha$, then
$A( 1 ) A(4)+A(2) A(5)=$

TS EAMCET 2018

Solution:

$\because A(n)=\sin ^{n} \alpha+\cos ^{n} \alpha$
Then, $A(1) A(4)+A(2) A(5)$
$=(\sin \alpha+\cos \alpha)\left(\sin ^{4} \alpha+\cos ^{4} \alpha\right)$
$+\left(\sin ^{2} \alpha+\cos ^{2} \alpha\right)\left(\sin ^{5} \alpha+\cos ^{5} \alpha\right)$

$=(\sin \alpha+\cos \alpha)\left[\left(\sin ^{2} \alpha+\cos ^{2} \alpha\right)^{2}\right.$
$ \left.-2 \sin ^{2} \alpha \cos ^{2} \alpha\right]+\left(\sin ^{5} \alpha+\cos ^{5} \alpha\right) $
$=(\sin \alpha+\cos \alpha)\left[\left(\sin ^{2} \alpha+\cos ^{2} \alpha\right)^{3}-2 \sin ^{2} \alpha \cos ^{2} \alpha\right] $
$+\left(\sin ^{5} \alpha+\cos ^{5} \alpha\right) $
$=(\sin \alpha+\cos \alpha)\left[\sin ^{6} \alpha+\cos ^{6} \alpha+\sin ^{2} \alpha \cos ^{2} \alpha\right] $
$ +\left(\sin ^{5} \alpha+\cos ^{5} \alpha\right) $
$=(\sin \alpha+\cos \alpha)\left(\sin ^{6} \alpha+\cos ^{6} \alpha\right) $
$+\sin ^{2} \alpha \cos ^{2} \alpha(\sin \alpha+\cos \alpha)+\left(\sin ^{5} \alpha+\cos ^{5} \alpha\right) $
$=(\sin \alpha+\cos \alpha)\left(\sin ^{6} \alpha+\cos ^{6} \alpha\right) $
$+\left(\sin ^{3} \alpha \cos ^{2} \alpha+\sin ^{5} \alpha\right)+\left(\sin ^{2} \alpha \cos ^{3} \alpha+\cos ^{5} \alpha\right)$
$=(\sin \alpha+\cos \alpha)\left(\sin ^{6} \alpha+\cos ^{6} \alpha\right) $
$+\left(\sin ^{2} \alpha+\cos ^{2} \alpha\right)\left(\sin ^{3} \alpha+\cos ^{3} \alpha\right)$
$= A(1) A(6)+A(2) A(3)$