Use the product-to-sum formula to find the exact value: If cos37.5∘sin7.5∘=(√A−B)/4, then A= B=

Question

please provide all steps for the solution

Answer 1

The product identity states that:

sin(a)cos(b) = 0.5[sin(a+b) + sin(a-b)]

Using this identity, we can plug in "a" and "b":

sin(7.5°)cos(37.5°) = 0.5[sin(7.5° + 37.5°) + sin(7.5° - 37.5°)]

= 0.5[sin(45°) + sin(-30°)]

= 0.5[√(2)/2 - sin(30°)] , Sine is odd

= 0.5[√(2)/2 - 1/2]

= [√(2) - 1]/4

Therefore, A = 2, B = 1.