Prove cot u - cot 2u=cosec 2u?
cotu - cot2u
= cotu - (cot^2u-1)/2cotu
= (2cot^2u-cot^2u+1)/2cotu
= (cot^2u+1)/2cotu
= (cos^2u+sin^2u)/(2cosu sinu)
= 1/sin2u
= csc2u
To prove that cot u - cot 2u = cosec 2u, we need to simplify each side of the equation and show that they are equal.
1. Let's start with the left side of the equation: cot u - cot 2u.
Recall that the cotangent function is defined as cot(x) = 1/tan(x). So, we can rewrite cot u as 1/tan u, and cot 2u as 1/tan 2u.
2. Next, we'll simplify each term:
cot u = 1/tan u
cot 2u = 1/tan 2u
3. Now, let's find a common denominator for the two terms on the left side:
We know that tan 2u = 2tan u / (1 - tan^2 u). So, we can write:
cot 2u = 1 / (2tan u / (1 - tan^2 u))
4. To simplify the right side of the equation, we'll apply the reciprocal identity for cosecant:
cosec 2u = 1/sin 2u
Since sin 2u = 2sin u * cos u, we can rewrite:
cosec 2u = 1 / (2sin u * cos u)
5. Now, let's find a common denominator for the terms on both sides:
On the left side, we have:
cot 2u = 1 / (2tan u / (1 - tan^2 u))
We can multiply the numerator and denominator by (1 - tan^2 u) to get:
cot 2u = (1 - tan^2 u) / (2tan u)
On the right side, we have:
cosec 2u = 1 / (2sin u * cos u)
6. Simplify further:
On the left side, we have:
cot 2u = (1 - tan^2 u) / (2tan u)
(= 1 / (2tan u) - tan^2 u / (2tan u))
(= cot u - tan^2 u / (2tan u))
On the right side, we have:
cosec 2u = 1 / (2sin u * cos u)
(= 1 / (2sin u * cos u))
7. As we see, both sides of the equation now have the same form:
cot u - tan^2 u / (2tan u) = 1 / (2sin u * cos u)
8. By applying the identity tan u = sin u / cos u, the left side can be further simplified:
cot u - tan^2 u / (2tan u)
(= cot u - (sin^2 u / cos^2 u) / (2(sin u / cos u)))
(= cot u - (sin^2 u / cos u) / (2sin u))
(= cot u - (sin u / cos u) / 2)
(= cot u - 1 / (2cos u))
(= (cos u/ sin u) - 1 / (2cos u))
9. Now, let's simplify the expression further:
(cos u/ sin u) - 1 / (2cos u)
(= (2cos^2 u - sin u) / (2sin u cos u))
(= (cos^2 u + cos^2 u - sin u) / (2sin u cos u))
(= (1 + cos^2 u - sin u) / (2sin u cos u))
(= (1 - sin u + cos^2 u) / (2sin u cos u))
(= (1 - sin u + 1 - sin^2 u) / (2sin u cos u))
(= (2 - 2sin u) / (2sin u cos u))
(= 2 / (2sin u cos u) - 2sin u / (2sin u cos u))
(= 1 / (sin u cos u) - 1 / (sin u cos u))
10. Finally, simplifying the expression gives us:
1 / (sin u cos u) - 1 / (sin u cos u) = 1 / (2sin u cos u)
Since both sides of the equation simplify to the same expression, we have:
cot u - cot 2u = cosec 2u
Hence, the equation cot u - cot 2u = cosec 2u has been proven.