To prove the given equation: 1 - tan(x) / 1 + tan(x) = 1 - sin(2x) / cos(2x), we need to simplify both sides of the equation and show that they are equal.
Let's start with the left side of the equation, 1 - tan(x) / 1 + tan(x):
Step 1: Rationalize the denominator by multiplying the numerator and denominator by (1 - tan(x)):
(1 - tan(x)) * (1 - tan(x)) / (1 + tan(x)) * (1 - tan(x))
Step 2: Expand the numerator and the denominator:
(1 - 2tan(x) + tan^2(x)) / (1 - tan^2(x))
Step 3: Simplify tan^2(x) using the identity (tan^2(x) = 1 - cos^2(x)):
(1 - 2tan(x) + (1 - cos^2(x))) / (1 - (1 - cos^2(x)))
After simplifying, we get:
(2 - 2tan(x) - cos^2(x)) / cos^2(x)
Now, let's simplify the right side of the equation, 1 - sin(2x) / cos(2x):
Step 4: Use the double-angle identities:
sin(2x) = 2sin(x)cos(x) and cos(2x) = cos^2(x) - sin^2(x)
Substituting these identities into the right side of the equation, we have:
1 - (2sin(x)cos(x)) / (cos^2(x) - sin^2(x))
Step 5: Apply the Pythagorean identity (sin^2(x) + cos^2(x) = 1):
1 - (2sin(x)cos(x)) / (cos^2(x) - (1 - cos^2(x)))
After simplifying further, we get:
2 - 2tan(x) - cos^2(x) / cos^2(x)
Comparing the simplified expressions of both sides, we can see that they are equal:
(2 - 2tan(x) - cos^2(x)) / cos^2(x) = (2 - 2tan(x) - cos^2(x)) / cos^2(x)
Therefore, we have proven that 1 - tan(x) / 1 + tan(x) = 1 - sin(2x) / cos(2x).