How do I prove that sin(x+y)-sin(x-y)=2cosx siny
just apply the addition formulas:
sin(x+y) = sinx cosy + cosx siny
sin(x-y) = sinx cosy - cosx siny
now subtract. done.
or, apply the sum-to-product formula directly: sinA-sinB = 2sin((A-B)/2)cos((A+B)/2)
so, letting A=x+y and B=x-y
sin(x+y)-sin(x-y) = 2sin(((x+y)-(x-y))/2) cos(((x+y)+(x-y))/2)
= 2 siny cosx
but then, the sum-to-product formula is just based on the addition formulas anyway, so what'd you expect?
To prove the identity sin(x+y) - sin(x-y) = 2cos(x)sin(y), we will use the trigonometric identities and the properties of sine and cosine functions.
Let's start with the left-hand side of the equation:
sin(x+y) - sin(x-y)
Using the sum-to-product identity for sine, we can rewrite sin(x+y) as sin(x)cos(y) + cos(x)sin(y).
Similarly, sin(x-y) can be written as sin(x)cos(y) - cos(x)sin(y).
Replacing sin(x+y) and sin(x-y) in the original equation, we obtain:
(sin(x)cos(y) + cos(x)sin(y)) - (sin(x)cos(y) - cos(x)sin(y))
Now, let's simplify the expression:
sin(x)cos(y) + cos(x)sin(y) - sin(x)cos(y) + cos(x)sin(y)
The terms sin(x)cos(y) and -sin(x)cos(y) cancel each other out.
Simplifying further, we have:
cos(x)sin(y) + cos(x)sin(y)
Combining like terms, we get:
2cos(x)sin(y)
Therefore, we have proved that sin(x+y) - sin(x-y) = 2cos(x)sin(y).
To summarize the steps:
1. Use the sum-to-product identity for sine to express sin(x+y) and sin(x-y) in terms of sine and cosine functions.
2. Substitute these expressions back into the original equation.
3. Simplify by combining like terms and canceling out common terms.
4. The final result should match the right-hand side of the equation, which is 2cos(x)sin(y).
To prove the identity sin(x+y) - sin(x-y) = 2cos(x)sin(y), you can use the sum-to-product trigonometric identity. Here's a step-by-step proof:
Step 1: Start with the left-hand side of the equation:
sin(x+y) - sin(x-y)
Step 2: Expand the first term using the sum-to-product identity sin(u+v) = sin(u)cos(v) + cos(u)sin(v):
(sin(x)cos(y) + cos(x)sin(y)) - sin(x-y)
Step 3: Distribute the negative sign to the second term:
sin(x)cos(y) + cos(x)sin(y) - sin(x) + sin(y)
Step 4: Rearrange the terms to group the sine and cosine terms:
(sin(x) - sin(x)) + (cos(x)sin(y) + sin(y)cos(y))
Step 5: Notice that the first term cancels out:
0 + (cos(x)sin(y) + sin(y)cos(y))
Step 6: Combine the like terms:
cos(x)sin(y) + sin(y)cos(y)
Step 7: Use the double-angle identity sin(2θ) = 2sin(θ)cos(θ) to simplify further:
2cos(x)sin(y) (since cos(y) = sin(y) using the symmetry of sine and cosine functions)
Therefore, we've proved that sin(x+y) - sin(x-y) = 2cos(x)sin(y).