Please Help:
please solve this proof:
cosx- cosy = -2sin(x+y/2)sin(x-y/2)
let (x+y)/2 = A ---> x+y=2A
let (x-y)/2 = B ---> x-y=2B
adding those : 2x = 2A + 2B
then x = A+B and similarly
y = A - B
so your identity becomes
cos(A+B) - cos(A-B) = -2sinAsinB
LS
= cosAcosB - sinAsinB -(cosAcosB + sinAsinB)
= -2sinAsinB
= -2sin(x+y/2)sin(x-y/2)
= RS
Thanks so much for the help=)
To prove that cos(x) - cos(y) = -2sin((x+y)/2)sin((x-y)/2), we will utilize the trigonometric identity:
cos(A) - cos(B) = -2sin((A+B)/2)sin((A-B)/2).
Comparing the identity with the given equation, we can see that A = x and B = y. Therefore, we need to show that:
cos(x) - cos(y) = -2sin((x+y)/2)sin((x-y)/2).
To prove this, we will start with the left-hand side of the equation, cos(x) - cos(y), and attempt to simplify it to the right-hand side. We will use the sum-to-product identities for sine and cosine to manipulate the equation step by step:
Step 1: Start with cos(x) - cos(y).
Step 2: Rewrite cos(x) as sin(π/2 - x) using the complementary angle identity.
Step 3: Substitute this into the equation: sin(π/2 - x) - cos(y).
Step 4: Apply the sum-to-product identity for sine: sin(A) - sin(B) = -2cos((A+B)/2)sin((A-B)/2).
Step 5: Rewrite the equation as -2cos((y + π/2 - x)/2)sin((y - π/2 + x)/2).
Step 6: Simplify the angles in the cos and sin functions:
-2cos((y + π/2 - x)/2)sin((y - π/2 + x)/2)
= -2cos((π/2 + y - x)/2)sin((y - π/2 + x)/2)
= -2cos((π/4 + y/2 - x/2))sin((y - π/2 + x)/2)
= -2sin((x + y)/2)sin((y - x)/2).
Step 7: Simplify the sign: -2sin((x + y)/2)sin((y - x)/2) = -2sin((x + y)/2)sin((x - y)/2).
Thus, we have proven the given equation cos(x) - cos(y) = -2sin((x+y)/2)sin((x-y)/2) using the trigonometric identity cos(A) - cos(B) = -2sin((A+B)/2)sin((A-B)/2).