The value of the following expression is 3(si | vedclass

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(D) To find the value of the expression,we can substitute a convenient value for $	heta$. Let $	heta = 0^{\circ}$.Since $\sin 0^{\circ} = 0$ and $\c

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$5$

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(D) To find the value of the expression,we can substitute a convenient value for $	heta$. Let $	heta = 0^{\circ}$.Since $\sin 0^{\circ} = 0$ and $\cos 0^{\circ} = 1$,we substitute these into the expression:$= 3(0^{4} + 1^{4}) + 2(0^{6} + 1^{6}) + 12(0^{2})(1^{2})$$= 3(0 + 1) + 2(0 + 1) + 12(0)(1)$$= 3(1) + 2(1) + 0$$= 3 + 2 = 5$.Alternatively,using identities:$\sin^{4} 	heta + \cos^{4} 	heta = 1 - 2\sin^{2} 	heta \cos^{2} 	heta$$\sin^{6} 	heta + \cos^{6} 	heta = 1 - 3\sin^{2} 	heta \cos^{2} 	heta$Substituting these:$= 3(1 - 2\sin^{2} 	heta \cos^{2} 	heta) + 2(1 - 3\sin^{2} 	heta \cos^{2} 	heta) + 12\sin^{2} 	heta \cos^{2} 	heta$$= 3 - 6\sin^{2} 	heta \cos^{2} 	heta + 2 - 6\sin^{2} 	heta \cos^{2} 	heta + 12\sin^{2} 	heta \cos^{2} 	heta$$= 3 + 2 + (-6 - 6 + 12)\sin^{2} 	heta \cos^{2} 	heta$$= 5 + 0 = 5$.

The value of the following expression is $3(\sin^{4} \theta + \cos^{4} \theta) + 2(\sin^{6} \theta + \cos^{6} \theta) + 12\sin^{2} \theta \cos^{2} \theta$.

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