COMEDK 2024 Shift 1 Paper

We can use the product-to-sum formula:
cos‌A‌cos‌B=‌

1

2

[cos(A+B)+cos(A−B)]
to simplify the given expression. Let's start by pairing the first and fourth terms, and the second and third terms:
(1+cos‌

π

8

)(1+cos‌

7π

8

)
‌=1+cos‌

π

8

+cos‌

7π

8

+‌

1

2

[−1−‌

1

√2

]
‌=‌

1

2

+cos‌

π

8

+cos‌

7π

8

−‌

1

2√2

‌=‌

1

2

+cos‌

π

8

+cos(π−‌

π

8

)−‌

1

2√2

‌=‌

1

2

+cos‌

π

8

−cos‌

π

8

−‌

1

2√2

‌=‌

1

2

−‌

1

2√2

‌=‌

√2−1

2√2

Similarly, for the second and third terms:
(1+cos‌

3π

8

)(1+cos‌

5π

8

)
‌=1+cos‌

3π

8

+cos‌

5π

8

+‌

1

2

[−1+‌

1

√2

]
‌=‌

1

2

+cos‌

3π

8

+cos‌

5π

8

+‌

1

2√2

‌=‌

1

2

+cos‌

3π

8

+cos(π−‌

3π

8

)+‌

1

2√2

‌=‌

1

2

+cos‌

3π

8

−cos‌

3π

8

+‌

1

2√2

‌=‌

1

2

+‌

1

2√2

‌=‌

√2+1

2√2

Now, multiplying these two simplified expressions:
‌4(‌

√2−1

2√2

)(‌

√2+1

2√2

)
‌=4(‌

(√2)2−12

4⋅2

)
‌=4(‌

1

8

)
‌=‌

1

2

Therefore, the answer is Option C.