Evaluate: sqrt{sin ^4 x+4 cos ^2 x}-sqrt{cos | vedclass

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(D) Given expression: $\sqrt{\sin ^4 x+4 \cos ^2 x}-\sqrt{\cos ^4 x+4 \sin ^2 x}$ Using the identity $\cos ^2 x = 1-\sin ^2 x$ and $\sin ^2 x = 1-\cos

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$\cos 2 x$

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(D) Given expression: $\sqrt{\sin ^4 x+4 \cos ^2 x}-\sqrt{\cos ^4 x+4 \sin ^2 x}$ Using the identity $\cos ^2 x = 1-\sin ^2 x$ and $\sin ^2 x = 1-\cos ^2 x$: $\sqrt{\sin ^4 x+4(1-\sin ^2 x)}-\sqrt{\cos ^4 x+4(1-\cos ^2 x)}$ $= \sqrt{\sin ^4 x-4 \sin ^2 x+4}-\sqrt{\cos ^4 x-4 \cos ^2 x+4}$ $= \sqrt{(\sin ^2 x-2)^2}-\sqrt{(\cos ^2 x-2)^2}$ Since $0 \le \sin ^2 x \le 1$ and $0 \le \cos ^2 x \le 1$,the terms inside the square roots are negative,so we take the absolute value: $= |\sin ^2 x-2|-|\cos ^2 x-2|$ $= (2-\sin ^2 x)-(2-\cos ^2 x)$ $= \cos ^2 x-\sin ^2 x$ $= \cos 2 x$

Evaluate: $\sqrt{\sin ^4 x+4 \cos ^2 x}-\sqrt{\cos ^4 x+4 \sin ^2 x}$

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