Answer:
Explanation:
To prove the identities using the sum and difference formulas, you can break down each identity step by step:
1. Proof for sin(π/2 + x) = cos(x):
Using the sum formula for sine, we have:
sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
In this case, let a = π/2 and b = x:
sin(π/2 + x) = sin(π/2)cos(x) + cos(π/2)sin(x)
Since sin(π/2) = 1 and cos(π/2) = 0, we can simplify further:
sin(π/2 + x) = 1 * cos(x) + 0 * sin(x)
sin(π/2 + x) = cos(x)
Therefore, we have proved that sin(π/2 + x) = cos(x) using the sum formula for sine.
2. Proof for sin(x+y) + sin(x-y) = 2sin(x)cos(y):
Using the sum formula for sine, we have:
sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
In this case, let a = x and b = y:
sin(x + y) = sin(x)cos(y) + cos(x)sin(y)
Similarly, using the difference formula for sine, we have:
sin(a - b) = sin(a)cos(b) - cos(a)sin(b)
In this case, let a = x and b = y:
sin(x - y) = sin(x)cos(y) - cos(x)sin(y)
Adding the two equations together, we get:
sin(x + y) + sin(x - y) = sin(x)cos(y) + cos(x)sin(y) + sin(x)cos(y) - cos(x)sin(y)
The terms sin(x)cos(y) and -cos(x)sin(y) cancel out:
sin(x + y) + sin(x - y) = 2sin(x)cos(y)
Therefore, we have proved that sin(x+y) + sin(x-y) = 2sin(x)cos(y) using the sum and difference formulas for sine.