Question

2sina-sin2a/cos2a-1 -ctga​

Answer 1

$\frac{2sina-sin2a}{cos2a-1}-ctga=\frac{2sina-2\, sina\, cosa}{(cos^2a-sin^2a)-(sin^2a+cos^2a)}-ctga=\frac{2\, sina\cdot (1-cosa)}{-2sin^2a}-ctga=\\\\=-\frac{1-cosa}{sina}-ctga=-\frac{2sin^2\frac{a}{2}}{2\, sin\frac{a}{2}\, cos\frac{a}{2}}-ctga=-\frac{sin\frac{a}{2}}{cos\frac{a}{2}}-ctga=-tg\frac{a}{2}-ctga\\\\ili\\\\...=-\frac{1-cosa}{sina}-\frac{cosa}{sina}=\frac{-1+cosa-cosa}{sina}=-\frac{1}{sina}$

Answer 2

To simplify the expression (2sin(a) - sin(2a))/(cos(2a) - 1) - ctg(a), we can start by manipulating the terms:

1. Recall that sin(2a) = 2sin(a)cos(a), so we can rewrite sin(2a) as 2sin(a)cos(a) in the numerator: (2sin(a) - 2sin(a)cos(a))/(cos(2a) - 1) - ctg(a)

2. Now, let's focus on the denominator, cos(2a) - 1: cos(2a) - 1 = cos^2(a) - sin^2(a) - 1 = (cos^2(a) - sin^2(a)) - 1 = cos^2(a) - (1 - sin^2(a)) - 1 = cos^2(a) - cos^2(a) - 1 = -1

   Therefore, the denominator simplifies to -1: (2sin(a) - 2sin(a)cos(a))/(-1) - ctg(a)

3. We can multiply the numerator by -1 to change the sign: (-2sin(a) + 2sin(a)cos(a)) - ctg(a)

4. Factor out 2sin(a) from the first two terms: 2sin(a)(1 - cos(a)) - ctg(a)

5. Finally, we can rearrange the terms: -ctg(a) + 2sin(a)(1 - cos(a))

So, the simplified expression is -ctg(a) + 2sin(a)(1 - cos(a)).