If ABCD is a cyclic quadrilateral, then: cos(180∘+A)+cos(180∘−B)+cos(180∘−C)−sin(90∘−D)=

A. -1
B. 0
C. 1
D. 2

We are given a cyclic quadrilateral ABCD, and we need to find the value of the expression:

cos(180° + A) + cos(180° - B) + cos(180° - C) - sin(90° - D).

Let's break it down step by step:

Step 1: Simplify each trigonometric term
cos(180° + A):

Using the identity cos(180° + θ) = -cos(θ), we get: cos(180° + A) = -cos(A).
cos(180° - B):

Using the identity cos(180° - θ) = -cos(θ), we get: cos(180° - B) = -cos(B).
cos(180° - C):

Again, using the identity cos(180° - θ) = -cos(θ), we get: cos(180° - C) = -cos(C).
sin(90° - D):

Using the identity sin(90° - θ) = cos(θ), we get: sin(90° - D) = cos(D).
Step 2: Substitute these simplifications into the original expression
Now, substitute the simplified terms into the original expression:

cos(180° + A) + cos(180° - B) + cos(180° - C) - sin(90° - D) = -cos(A) - cos(B) - cos(C) - cos(D).

Step 3: Interpret the result
For a cyclic quadrilateral, the angles A, B, C, and D satisfy the property:

A + B + C + D = 360° (since the sum of opposite angles in a cyclic quadrilateral is 180°).

However, the expression we've derived does not directly use this property, and we can look at the possible values for cos(A), cos(B), cos(C), and cos(D). Without specific values for the angles, the best way to proceed is to recognize that the sum of cosines of the angles typically results in zero in this case due to symmetry in a cyclic quadrilateral.

Step 4: Conclusion
Thus, the value of the expression is:

0.

The correct answer is B. 0.