Multiple choice

The value of $4(sin^{4} 30^{0} +cos^{4} 60^{0})(cos^{2} 45^{0} - sin^{2} 90^{0})+tan^{4}45^{0}-cot^{2}45^{0}$ is

  1. $0$
  2. $1$
  3. $-\dfrac{1}{4}$

  4. -2
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C Correct answer
Explanation

sin 30 = 1/2, cos 60 = 1/2, cos 45 = 1/sqrt(2), sin 90 = 1, tan 45 = 1, cot 45 = 1. Expression: 4((1/16) + (1/16)) * (1/2 - 1) + 1 - 1 = 4(2/16) * (-1/2) = 4(1/8) * (-1/2) = -1/4.

AI explanation

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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