Multiple and sub- multiple Angles

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Question

The value of $\cos^{2} A {(3 - 4 \cos^{2} A)}^{2} + \sin^{2} A {(3 - 4 \sin^{2} A)}^{2}$ is equal to

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Solution

$\begin{array}{l} \cos^{2} A {(3 - 4 \cos^{2} A)}^{2} + \sin^{2} A {(3 - 4 \sin^{2} A)}^{2} \\ = {(3 \cos A - 4 \cos^{3} A)}^{2} + {(3 \sin A - 4 \sin^{3} A)}^{2} \\ = (- \cos 3 A)^{2} + (\sin 3 A)^{2} = 1 \end{array}$

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