Verify the identity:
sin(x+y)/cos(x-y)=cotx+coty/1+cotxcoty
To verify the given identity, we need to simplify both sides and demonstrate that they are equal. Let's start by simplifying the left-hand side (LHS).
LHS: sin(x+y)/cos(x-y)
To simplify this expression, we'll use the identities for sine, cosine, and cotangent.
Recall the following trigonometric identities:
1. sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
2. cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
3. cot(A) = cos(A)/sin(A)
Applying identities 1 and 2 to the numerator and denominator of the LHS, we get:
LHS: (sin(x)cos(y) + cos(x)sin(y)) / (cos(x)cos(y) + sin(x)sin(y))
Now, let's simplify the right-hand side (RHS):
RHS: cot(x) + cot(y) / (1 + cot(x)cot(y))
Using identity 3 for cotangent, we can rewrite the RHS as:
RHS: (cos(x)/sin(x) + cos(y)/sin(y)) / (1 + (cos(x)/sin(x))(cos(y)/sin(y)))
Taking the common denominator for the numerator of the RHS and simplifying, we get:
RHS: (cos(x)sin(y) + cos(y)sin(x)) / (sin(x)sin(y) + cos(x)cos(y))
Now, to verify the identity, we need to show that LHS = RHS. Comparing the simplified expressions, we can see that LHS = RHS. Therefore, the given identity is true.
Hence, the identity sin(x+y)/cos(x-y) = cot(x) + cot(y) / (1 + cot(x)cot(y)) has been verified to be true.