Show that cot((x+y)/2) = - (sin x - sin y)/(cos x - cos y) for all values of x and y for which both sides are defined.
I tried manipulating both sides in terms of trig identities but I don't really have a solution....help would be appreciated, thanks.
Using the sum-to-product formulas,
sinx-siny = 2cos((x+y)/2)sin((x-y)/2)
cosx-cosy = -2sin((x+y)/2)sin((x-y)/2)
now just divide to get
cos((x+y)/2) / sin((x+y)/2) = cot((x+y)/2)
Two of the standard conversion formulas are:
sinA - sinB = 2sin((A-B)/2) cos( (A+B)/2)
and
cosA - cosB = - 2sin( (A-B)/2) sin( (A+B)/2)
see:
near bottom of page under:
Sum-to-Product Formulas
http://www.sosmath.com/trig/Trig5/trig5/trig5.html
RS
= -(sinx - siny)/(cosx - cosy)
= - 2sin((X-Y)/2) cos( (X+Y)/2)/- 2sin( (X-Y)/2) sin( (X+Y)/2)
= cos((X+Y)/2) / sin(X+Y)/2)
= cot((X+Y)/2)
= LS
To prove that cot((x+y)/2) = -(sin x - sin y)/(cos x - cos y), we need to manipulate both sides of the equation to be equivalent. Let's start with the left-hand side (LHS) and right-hand side (RHS) separately:
LHS: cot((x+y)/2)
Recall the following trigonometric identities:
1. cot θ = cos θ / sin θ
2. cos (a + b) = cos a cos b - sin a sin b
Using the first identity, we can rewrite cot((x+y)/2) as cos((x+y)/2) / sin((x+y)/2).
Now, let's focus on the RHS: -(sin x - sin y)/(cos x - cos y)
To simplify this, we can multiply the numerator and the denominator by (cos x + cos y), which is the conjugate of (cos x - cos y). This will help us simplify the expression later.
RHS: -(sin x - sin y)/(cos x - cos y) * (cos x + cos y)/(cos x + cos y)
Expanding the numerator and the denominator, we get:
RHS: -(sin x cos x - sin x cos y - sin y cos x + sin y cos y) / (cos x cos x - cos x cos y - cos y cos x + cos y cos y)
Simplifying the numerator and the denominator, we have:
RHS: -(sin x cos y - sin y cos x) / (cos x cos y - cos y cos x)
Notice that the numerator is the same as the numerator on the LHS, and the denominator is the negative of the denominator on the LHS.
Now, let's simplify the RHS further:
RHS: (sin y cos x - sin x cos y) / (cos y cos x - cos x cos y)
= (sin y cos x - sin x cos y) / (cos x cos y - cos x cos y)
= (sin y cos x - sin x cos y) / 0
Since the denominator is zero, this means that both sides are not defined for the same values of x and y. Therefore, we cannot conclude that the given equation is true for all values of x and y for which both sides are defined.
In conclusion, the equation cot((x+y)/2) = -(sin x - sin y)/(cos x - cos y) is not true for all values of x and y where both sides are defined.