Sin 2x / 2 sin x + cos2 x -1=1 + sign x / cosx

Answer 1

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.