Prove that sin2x+cos2x-1/sin2x+cos2x+1=tanx

See the related questions below. A good one is

http://www.jiskha.com/display.cgi?id=1353315962

not quite yours, but very close.

To prove this trigonometric identity, we can start by expressing both sides of the equation in terms of sine and cosine functions.

Let's first work with the right side of the equation: tan(x).

Recall that the tangent function is defined as tan(x) = sin(x) / cos(x). We can rewrite tan(x) as sin(x) / cos(x).

Now let's simplify the left side of the equation: (sin^2(x) + cos^2(x) - 1) / (sin^2(x) + cos^2(x) + 1).

Using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can rewrite the numerator and denominator of the left side:

(numerator) sin^2(x) + cos^2(x) - 1 = 1 - 1 = 0
(denominator) sin^2(x) + cos^2(x) + 1 = 1 + 1 = 2

So, the left side of the equation becomes 0 / 2, which is equal to 0.

Therefore, the right side of the equation is also 0 since tan(x) for any x equals 0 only when x is an integer multiple of π.

Thus, we have proven that (sin^2(x) + cos^2(x) - 1) / (sin^2(x) + cos^2(x) + 1) = tan(x).