We know the formula for cos(A - B):
cos(A - B) = cos(A) * cos(B) + sin(A) * sin(B)
Now, we'll introduce a new angle, C, such that A = (C + B). In other words, angle A is the sum of angles C and B.
Substituting A = C + B into the formula for cos(A - B):
cos((C + B) - B) = cos(C + B) = cos(C) * cos(B) + sin(C) * sin(B)
Now, we want to find the value of cos(C + B), which is essentially cos(A + B). We can rewrite this equation as:
cos(A + B) = cos(C) * cos(B) + sin(C) * sin(B)
So, by introducing a new angle C and utilizing the formula for cos(A - B), we have derived the compound angle formula for cos(A + B):
cos(A + B) = cos(A) * cos(B) - sin(A) * sin(B)
This is the formula for cos(A + B) derived from the formula for cos(A - B) using the concept of complementary angles.
Let's start with the formula for sin(A + B):
sin(A+B)=sin(A)∗cos(B)+cos(A)∗sin(B)
Now, we'll substitute the given information A=\(\frac{\pi}{2}\)−A, which implies A+A=\(\frac{\pi}{2}\):
sin(\(\frac{\pi}{2}\))=sin(A)∗cos(B)+cos(A)∗sin(B)
Since sin(\(\frac{\pi}{2}\))=1, the equation simplifies to:
1=sin(A)∗cos(B)+cos(A)∗sin(B)
Now, we'll use the formula for cos(A - B):
cos(A−B)=cos(A)∗cos(B)+sin(A)∗sin(B)
Substituting A=\(\frac{\pi}{2}\)−A into the cos(A - B) formula:
cos(\(\frac{\pi}{2}\)−A−B)=cos(\(\frac{\pi}{2}\)−A)∗cos(B)+sin(\(\frac{\pi}{2}\)−A)∗sin(B)
Using the trigonometric identity sin(\(\frac{\pi}{2}\)−x)=cos(x):
cos(\(\frac{\pi}{2}\)−A−B)=cos(A)∗cos(B)+cos(A)∗sin(B)
Simplifying further:
cos(\(\frac{\pi}{2}\)−A−B)=cos(A)∗(cos(B)+sin(B))
Since cos(\(\frac{\pi}{2}\)−x)=sin(x), we can rewrite the left side:
sin(A+B)=cos(A)∗(cos(B)+sin(B))
Now, using the equation 1=sin(A)∗cos(B)+cos(A)∗sin(B), we can solve for sin(A)∗cos(B):
sin(A)∗cos(B)=1−cos(A)∗sin(B)
Substituting this back into the previous equation:
sin(A+B)=cos(A)∗(1−cos(A)∗sin(B))
This is the expression for sin(A + B) in terms of cos(A - B) and the given relationship A=\(\frac{\pi}{2}\)−A.
When we replace B with −B, the expression for sin(A+B) in terms of cos(A−B) becomes:
sin(A−B)=cos(A)⋅(1−cos(A)⋅sin(−B))
Remember that sin(−x)=−sin(x), so sin(−B)=−sin(B). Substituting this into the equation:
sin(A−B)=cos(A)⋅(1−cos(A)⋅(−sin(B)))