Sin 2x / 2 sin x + cos2 x -1=1 + sign x / cosx
To simplify the given expression, we can start by working with each side separately.
Let's first simplify the left side of the equation:
sin 2x / 2 sin x + cos^2 x - 1
Using the double angle identity for sine (sin 2x = 2 sin x cos x), we can rewrite the expression as:
(2 sin x cos x) / 2 sin x + cos^2 x - 1
Next, simplify the expression by dividing both the numerator and denominator by sin x:
2 cos x / 2 + (cos^2 x - 1) = cos x + (cos^2 x - 1)
Now, simplify the expression (cos^2 x - 1) using the Pythagorean identity (cos^2 x = 1 - sin^2 x):
cos x + (1 - sin^2 x - 1) = cos x - sin^2 x
Now, let's move to simplify the right side of the equation:
1 + sin x / cos x
Now, find a common denominator for the expression:
(cos x / cos x) + sin x / cos x = (cos x + sin x) / cos x
Combining both sides of the equation, we get:
cos x - sin^2 x = (cos x + sin x) / cos x
Thus, the simplified form of the given expression is cos x - sin^2 x = (cos x + sin x) / cos x.