The value of $ \displaystyle 4(\sin ^{4}30^{\circ}+cos^{4}60^{\circ})(\cos ^{2}45^{\circ}-\sin ^{2}90^{\circ})+\tan ^{4}45^{\circ}-\cot ^{2}45^{\circ} $ is

Using standard trigonometric values, we substitute sin 30 = 1/2, cos 60 = 1/2, cos 45 = 1/2, sin 90 = 1, tan 45 = 1, and cot 45 = 1. The expression becomes 4((1/2)^4 + (1/2)^4)((1/2)^2 - 1^2) + 1^4 - 1^2, which simplifies to 4(1/16 + 1/16)(1/4 - 1) + 1 - 1. Working through the arithmetic gives 4(1/8)(-3/4) + 0, resulting in -1/4.

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