If sin A + sin 2A = x and cos A + cos 2A = y, | vedclass

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(A) Given: $x = \sin A + \sin 2A$ and $y = \cos A + \cos 2A$. Squaring and adding $x^2 + y^2$: $x^2 + y^2 = (\sin A + \sin 2A)^2 + (\cos A + \cos 2A)^

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$2y$

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(A) Given: $x = \sin A + \sin 2A$ and $y = \cos A + \cos 2A$. Squaring and adding $x^2 + y^2$: $x^2 + y^2 = (\sin A + \sin 2A)^2 + (\cos A + \cos 2A)^2$ $x^2 + y^2 = (\sin^2 A + \cos^2 A) + (\sin^2 2A + \cos^2 2A) + 2(\sin A \sin 2A + \cos A \cos 2A)$ $x^2 + y^2 = 1 + 1 + 2 \cos(2A - A)$ $x^2 + y^2 = 2 + 2 \cos A = 2(1 + \cos A)$. Now,substitute this into the expression $E = (x^2 + y^2)(x^2 + y^2 - 3)$: $E = [2(1 + \cos A)][2(1 + \cos A) - 3]$ $E = [2(1 + \cos A)][2 + 2 \cos A - 3]$ $E = [2(1 + \cos A)][2 \cos A - 1]$ $E = 2(2 \cos^2 A + 2 \cos A - \cos A - 1)$ $E = 2(2 \cos^2 A + \cos A - 1)$. Since $y = \cos A + \cos 2A = \cos A + (2 \cos^2 A - 1)$,we have $y = 2 \cos^2 A + \cos A - 1$. Therefore,$E = 2y$.

If $\sin A + \sin 2A = x$ and $\cos A + \cos 2A = y,$ then $({x^2} + {y^2})({x^2} + {y^2} - 3) = $

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