If $sin A + sin 2 A = x, cos A = x, cos A + cos 2 A = y,$
then prove that - $(x^{2} + y^{2}) (x^{2} + y^{2} - 3) = 2 y$ .

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Solution

Substituting values
$[(sin A + sin 2 A)^{2} + (cos A + cos 2 A)^{2}] [(sin A + sin 2 A)^{2} + (cos A + cos 2 A)^{2} - 3]$
$= [1 + 1 + cos A - cos 3 A + cos 3 A + cos A] [2 + 2 cos A - 3]$
$= (2 + 2 cos A) (2 cos A - 1)$
$= 4 cos A - 2 + 4 cos A - 2 cos A$
$= 4 {cos}^{2} A + 2 cos A - 2$
$= 2 (2 {cos}^{2} A - 1) + 2 cos A$
$= 2 cos 2 A + 2 cos A$
$= 2 (cos A + cos 2 A) = 2 y$
Hence proved.

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