If
sin‌A=‌ and
cos‌B=‌ where
A and
B lie in first and third quadrant
We know that:
cos(A+B)=cos‌A‌cos‌B−sin‌Asin‌BFirst, 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=‌‌cos2A+(‌)2=1‌cos2A=1−‌=‌ Therefore,
cos‌A=‌ (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=‌ :
‌(‌)2+sin‌2B=1‌sin‌2B=1−‌=‌Therefore,
sin‌B=‌ (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.