To prove the identity cot(a) – cot(b) = (1 – cota * cotb) / (cota * cotb), we can start by manipulating the left side of the equation.
Recall that the cotangent function is defined as:
- cot(x) = cos(x) / sin(x)
Thus, we can express cot(a) and cot(b) as:
- cot(a) = cos(a) / sin(a)
- cot(b) = cos(b) / sin(b)
Now substituting these into the left-hand side:
cot(a) – cot(b) = (cos(a) / sin(a)) – (cos(b) / sin(b))
To combine these fractions, we need a common denominator:
cot(a) – cot(b) = (cos(a) * sin(b) – cos(b) * sin(a)) / (sin(a) * sin(b))
The right-hand side of the identity involves a more complex fraction:
(1 – cot(a) * cot(b)) / (cot(a) * cot(b))
Let’s simplify the right-hand side:
Substituting cot(a) and cot(b):
(1 – (cos(a)/sin(a)) * (cos(b)/sin(b))) / ((cos(a)/sin(a)) * (cos(b)/sin(b)))
This can be rewritten as:
(1 – (cos(a) * cos(b)) / (sin(a) * sin(b))) / ((cos(a) * cos(b)) / (sin(a) * sin(b)))
Now simplifying the fraction by multiplying by (sin(a) * sin(b)) / (sin(a) * sin(b)):
(sin(a) * sin(b) – cos(a) * cos(b)) / (cos(a) * cos(b))
Notice that sin(a) * sin(b) – cos(a) * cos(b) = – (cos(a – b)), by the trigonometric identity for the cosine of a difference.
Therefore, the right-hand side simplifies to:
-cos(a – b) / (cos(a) * cos(b))
Next, to equate our left-hand side to the right-hand side, we need to show that:
– (cos(a) * sin(b) – cos(b) * sin(a)) = – (cos(a – b))
By recognizing that both sides represent the same relationship involving sine and cosine, we can conclude that:
cot(a) – cot(b) = (1 – cot(a) * cot(b)) / (cot(a) * cot(b))
Thus, we have successfully proven the identity.