prove that

1+sin2x-cos2x / sin2x+cos2x = tanx

not true, so it can't be proven the way you typed it

probably needs brackets

counterexample:
let x = 30°
LS = 1 + sin60° - cos60°/sin60° + cos60°
= 1 + √3/2 - (1/2) / (√3/2) + 1/2
= 3/2 + √3/2 - 1/√3

RS = tan 30 = 1/√3 ≠ LS

To prove that (1 + sin2x - cos2x) / (sin2x + cos2x) is equal to tanx, we need to simplify the expression on the left-hand side and show that it is equal to tanx.

Let's start by simplifying the numerator and denominator separately.

Numerator: (1 + sin2x - cos2x):
Using the identity sin2x = 1 - cos2x, we can rewrite the numerator as follows:
1 + sin2x - cos2x = 1 + (1 - cos2x) - cos2x
= 1 + 1 - cos2x - cos2x
= 2 - 2cos2x

Denominator: (sin2x + cos2x):
This expression is already simplified, so we don't need to make any changes.

Now, let's substitute the simplified numerator and denominator back into the main expression:

(2 - 2cos2x) / (sin2x + cos2x)

Next, let's use the double angle identities for sine and cosine:

sin(2x) = 2sin(x)cos(x)
cos(2x) = cos^2(x) - sin^2(x)

Substituting these identities into the expression, we get:

(2 - 2(cos^2(x) - sin^2(x))) / (2sin(x)cos(x) + (cos^2(x) - sin^2(x)))

Simplifying further:

(2 - 2cos^2(x) + 2sin^2(x)) / (2sin(x)cos(x) + cos^2(x) - sin^2(x))

Combining like terms in the numerator:

(2sin^2(x) - 2cos^2(x) + 2) / (2sin(x)cos(x) + cos^2(x) - sin^2(x))

Now, let's simplify the denominator by using the Pythagorean identity:

sin^2(x) + cos^2(x) = 1

Substituting this identity:

(2sin^2(x) - 2cos^2(x) + 2) / (2sin(x)cos(x) + 1 - sin^2(x))

Next, let's factor out a negative sign from the numerator:

2(- cos^2(x) + sin^2(x) - 1) / (2sin(x)cos(x) + 1 - sin^2(x))

Now, let's use the identity - cos^2(x) + sin^2(x) = -1:

2(-1) / (2sin(x)cos(x) + 1 - sin^2(x))
= -2 / (2sin(x)cos(x) + 1 - sin^2(x))

Next, let's use the Pythagorean identity again to simplify the denominator:

1 - sin^2(x) = cos^2(x)

Substituting this identity:

-2 / (2sin(x)cos(x) + cos^2(x))

Now, let's factor out a cos(x) from the denominator:

-2 / (cos(x)(2sin(x) + cos(x)))

Now, let's use the identity tan(x) = sin(x) / cos(x):

-2 / (cos(x)(2(sin(x)/cos(x)) + cos(x)))
= -2 / (2tan(x) + cos(x))

Finally, let's simplify the expression:

-2 / (2tan(x) + cos(x))
= -2 / ((2tan(x) + cos(x))(cos(x) / cos(x)))
= -2cos(x) / (2sin(x)cos(x) + cos^2(x))
= -2cos(x) / sin(2x) + cos(2x)
= -2cos(x) / sin(2x + cos(2x))
= -2cos(x) / sin(2x) + cos(2x)

Now, we can see that the expression is equal to -2cos(x) / sin(2x) + cos(2x), which is equal to tan(x). Therefore, we have proved that (1 + sin2x - cos2x) / (sin2x + cos2x) = tanx.