Verified answer

D. (-√(3)-1)/(2√(2))

It's sometimes easier to think in terms of degrees.

11pi/12 = 165 degrees; two angles we're familiar with are 120 and 45, so write

cos(165) = cos(120 + 45) = cos120cos45 - sin120sin45

cos(120) = -1/2; cos(45)= sin(45) = sqrt(2)/2; sin120 = sqrt(3)/2

cos(165) = (-1/2)(sqrt(2)/2) - (sqrt(3)/2)(sqrt(2)/2)

= -sqrt(2)/4 - sqrt(6)/4 = (-sqrt(2) - sqrt(6))/4

This is mathematically equivalent to answer D (multiply numerator and denominator by sqrt(2)) although it's not the form I'd prefer to see.

sin 15 = sin (40 5 - 30) distinction formula sin (A - B) = sin A cos B - cos A sin B sin 15 = sin 40 5 cos 30 - cos 40 5 sin 30 sin 15 = ((sqrt 2)/2)((sqrt 3)/2) - ((sqrt 2)/2) (a million/2) sin 15 = (sqrt 6)/4 - (sqrt 2)/4 sin 15 = (sqrt 6 - sqrt 2)/4

cos(11π/12) = cos(3π/12 + 8π/12)

= cos(π/4 + 2π/3)

= cos(π/4)cos(2π/3) - sin(π/4)sin(2π/3)

= (√2/2)(-1/2) - (√2/2)(√3/2)

= (-√2/4) - (√6/4)

= (-√2 - √6) / 4

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Now this does not match any of your answers.

So multiply numerator and denominator by √2:

= √2 (-√2 - √6) / (4√2)

= (-2 - 2√3) / (4√2)

= (-1 - √3) / (2√2)

ANSWER: D

You can use

11π/12 = 2π/12 + 9π/12 = π/6 + 3π/4.

=cos(pi/6 +3pi/4)=

cospi/6cos3pi/4 -sinpi/6sin3pi/4=

sqrt3/2(-sqrt2/2) -1/2(sqrt2/2) =

-sqrt6/4-sqrt2/4 is tha same as C