Integral of 1/(1-cos(2x)) (trigonometric identities)

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Integral of 1/(1-cos(2x)) (trigonometric identities)
Problem:
∫ 1 / (1 - cos(2x)) dx

Step 1: Use the Double-Angle Identity for Cosine
We know that:
cos(2x) = 1 - 2 sin^2(x)

Substitute this into the integral:
∫ 1 / (1 - (1 - 2 sin^2(x))) dx

Step 2: Simplify the Expression
Simplify the denominator:
∫ 1 / (2 sin^2(x)) dx

Step 3: Rewrite the Integral
Now, we can factor out the constant:
1/2 ∫ 1 / sin^2(x) dx

Step 4: Use the Cosecant Identity
We know that:
1 / sin^2(x) = csc^2(x)

So the integral becomes:
(1/2) ∫ csc^2(x) dx

Step 5: Integrate
The integral of csc^2(x) is -cot(x), so:
(1/2) (-cot(x)) + C

Final Answer:
-(1/2) cot(x) + C

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