Question

Ctg a * sin a/1-(sin a + cos a)^2

Answer 1

$ctg\alpha*\frac{sin\alpha}{1-(sin\alpha+cos\alpha)^2} = \frac{cos\alpha}{sin\alpha}*\frac{sin\alpha}{1 - (1+2sin\alpha cos\alpha)} = -\frac{cos\alpha}{2sin\alpha cos\alpha} = -\frac1{2sin\alpha}$

Answer 2

To simplify the expression, let's break it down step by step:

The given expression is: ctg(a) * sin(a) / [1 - (sin(a) + cos(a))^2]

Step 1: Calculate (sin(a) + cos(a))^2 (sin(a) + cos(a))^2 = sin^2(a) + 2sin(a)cos(a) + cos^2(a)

Step 2: Simplify 1 - (sin(a) + cos(a))^2 1 - (sin(a) + cos(a))^2 = 1 - (sin^2(a) + 2sin(a)cos(a) + cos^2(a)) = 1 - (sin^2(a) + cos^2(a)) - 2sin(a)cos(a) = 1 - 1 - 2sin(a)cos(a) (Using sin^2(a) + cos^2(a) = 1) = -2sin(a)cos(a)

Step 3: Substitute back into the original expression ctg(a) * sin(a) / [1 - (sin(a) + cos(a))^2] = ctg(a) * sin(a) / [-2sin(a)cos(a)]

Step 4: Cancel out common factors Since ctg(a) = 1/tan(a) = cos(a)/sin(a), we can simplify further: = (cos(a)/sin(a)) * sin(a) / [-2sin(a)cos(a)]

Step 5: Cancel out sin(a) terms = cos(a) / [-2cos(a)] = -1/2

So, the simplified expression is -1/2.