COMEDK 2024 Shift 1 Paper

If sin‌A=‌

4

5

and cos‌B=‌

−12

13

where A and B lie in first and third quadrant
We know that:

cos(A+B)=cos‌A‌cos‌B−sin‌Asin‌B
First, we need to find the values of cos‌A and sin‌B.
Since A lies in the first quadrant, both cos‌A and sin‌A are positive. We can use the Pythagorean identity to find cos‌A :
‌cos2A+sin‌2A=1
‌‌ Substituting ‌sin‌A=‌

4

5

‌cos2A+(‌

4

5

)2=1
‌cos2A=1−‌

16

25

=‌

9

25

Therefore, cos‌A=‌

3

5

(since cos‌A is positive in the first quadrant).
Since B lies in the third quadrant, both cos‌B and sin‌B are negative. We can again use the Pythagorean identity to find sin‌B :
cos2B+sin‌2B=1

Substituting cos‌B=‌

−12

13

:
‌(‌

−12

13

)2+sin‌2B=1
‌sin‌2B=1−‌

144

169

=‌

25

169

Therefore, sin‌B=‌

−5

13

(since sin‌B is negative in the third quadrant).
Now we can substitute all the values into the formula for cos(A+B) :
Therefore, the correct answer is Option D.