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Q.

If ABCD is a cyclic quadrilateral, then the value of cos A + cos B + cos C + cos D is:


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a

1

b

0

c

2

d

None of these 

answer is B.

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Detailed Solution

Step 1: First of all we have to consider the diagram of a cyclic quadrilateral ABCD which is as mentioned below:
seoStep 2: In cyclic quadrilateral the sum of its two opposite angles are equal to 1800 or π. Hence, we have to consider the two opposite angles A and C,
⇒ A + C = π
⇒ A = π – C .................(1)
Step 3: Multiplying with cosine on both sides of expression (1), we get
⇒ cosA = cos (π−C) ...............(2) Step 4: Now, to solve the expression (2)
⇒ cos A = cosπcosC + sinπsinC 
Now, on substituting the values of cosπ = -1, and sinπ = 0
⇒ cosA = (−1) cosC + (0) sinC
⇒ cosA = − cosC + 0
⇒ CosA + cosC = 0 ............(3)
Step 5: Now, consider other two opposite angles B and D,
⇒ B + D = π
⇒ B = π−D .................(4)
Step 6: Multiplying with cosine on both sides of expression (4), we get
⇒ cosB = cos(π−D) ...............(5)
Step 7: Now, to solve the expression (5)
⇒ cosB = cosπcosD + sinπsinD               [since cos (A − B) = cosAcosB + sinAsinB]
Now, on substituting the values cosπ = -1, and sinπ = 0, we get
⇒ cosB = (−1) cosD + (0) sinD
⇒ cosB = − cosD + 0
⇒ cosB + cosD = 0 ............(6)
Step 8: Now, we have to add the expressions (3) and (6) to obtain the value of cosA + cosB + cosC + cosD. Hence,
⇒ cosA + cosB + cosC + cosD = 0 + 0
⇒ cosA + cosB + cosC + cosD = 0
Therefore option (2) is correct.
 
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