Question

Sinx+siny+sin(x+y)=4sinx/2cosx/2cosx-y/2

Answer 1

sinx+siny+sin(x-y)=4sin(x/2)cos(x/2)cos((x-y)/2);

sinx+siny+sin(x-y)=2sin((x+y)/2)·cos((x-y)/2)+2sin(x-y)/2·cos(x-y)/2=

=2cos((x-y)/2)·[sin(x+y)/2+sin(x-y)/2)]=

=2cos((x-y)/2)·2sin((x+y+x-y)/4)·cos((x+y-x+y)/4)=

=4cos((x-y)/2)·sin(x/2)·cos(y/2)

Answer 2

It seems like you have an equation involving trigonometric functions that you want to simplify. Let's start by working on the left-hand side (LHS) of the equation and see if we can simplify it step by step.

Given equation: Sinx + Siny + Sin(x + y) = 4sinx/2cosx/2cosx - y/2

Step 1: Apply the sum-to-product trigonometric identities. The sum-to-product identities state:

1. sin(a) + sin(b) = 2 * sin((a + b) / 2) * cos((a - b) / 2)
2. sin(a) - sin(b) = 2 * cos((a + b) / 2) * sin((a - b) / 2)

Using the first identity, we can rewrite the equation as follows: 2 * sin((x + y) / 2) * cos((x - y) / 2) + Sin(x + y) = 4sinx/2cosx/2cosx - y/2

Step 2: Now, let's work on the right-hand side (RHS). The RHS looks a bit confusing, and it seems like there might be some confusion with the notation. Assuming you intended to represent the RHS as follows: 4 * sin(x/2) * cos(x/2) * cos((x - y) / 2)

Step 3: Apply double-angle trigonometric identities. The double-angle identities state:

1. sin(2a) = 2 * sin(a) * cos(a)
2. cos(2a) = cos^2(a) - sin^2(a) = 2 * cos^2(a) - 1 = 1 - 2 * sin^2(a)

Using the double-angle identities, we can simplify the RHS as follows: 4 * sin(x/2) * cos(x/2) * cos((x - y) / 2) = 2 * 2 * sin(x/2) * cos(x/2) * cos((x - y) / 2) = 2 * sin(x) * cos((x - y) / 2)

Step 4: Substitute the simplified RHS back into the equation. We have: 2 * sin((x + y) / 2) * cos((x - y) / 2) + Sin(x + y) = 2 * sin(x) * cos((x - y) / 2)

However, it seems there might still be some mistakes or typos in the equation or simplifications you provided. Double-check the original equation, or if you meant something different, please provide the correct equation so that I can assist you further.