Sin2x/1-Cos2x
You need two identities:
sin2A = sinAcosA and
cos2A = 1 - 2sin^2A
so Sin2x/(1-Cos2x)
= 2sinxcosx/(1 - (1-2sin^2x))
= 2sinxcosx/(2sin^2x)
= cosx/sinx
= cotx
To simplify the expression sin(2x)/(1 - cos(2x)), we can use trigonometric identities to rewrite it in terms of a single trigonometric function.
Step 1: Recall the Pythagorean identity sin^2(x) + cos^2(x) = 1.
Step 2: Using the double angle identity sin(2x) = 2sin(x)cos(x) and cos(2x) = cos^2(x) - sin^2(x), we can substitute these values in the expression.
sin(2x)/(1 - cos(2x)) = 2sin(x)cos(x) / (1 - (cos^2(x) - sin^2(x)))
Step 3: Simplify the denominator using the difference of squares identity a^2 - b^2 = (a + b)(a - b).
sin(2x)/(1 - cos(2x)) = 2sin(x)cos(x) / (1 - cos^2(x) + sin^2(x))
Step 4: Apply the Pythagorean identity sin^2(x) + cos^2(x) = 1 again to simplify the denominator further.
sin(2x)/(1 - cos(2x)) = 2sin(x)cos(x) / (sin^2(x) + sin^2(x))
Step 5: Combine the terms in the denominator.
sin(2x)/(1 - cos(2x)) = 2sin(x)cos(x) / (2sin^2(x))
Step 6: Simplify further by canceling out the common factor of 2.
sin(2x)/(1 - cos(2x)) = sin(x)cos(x) / sin^2(x)
Step 7: Finally, eliminate the common factor of sin(x) in the numerator and denominator.
sin(2x)/(1 - cos(2x)) = cos(x) / sin(x)
Therefore, sin(2x)/(1 - cos(2x)) simplifies to cos(x) / sin(x) or cot(x).