Ask AI: Prove Cos2×=1-tan²×÷1+tan²×

To prove that cos(2x) = (1 - tan^2(x))/(1 + tan^2(x)), we'll start by using the double angle identities and the definition of tangent. The double angle identity for cosine is: cos(2x) = 1 - 2*sin^2(x) Since tan(x) = sin(x)/cos(x), we can rewrite the expression (1 - tan^2(x))/(1 + tan^2(x)) in terms of sine and cosine. First, let's write tan^2(x) as (sin^2(x)/cos^2(x)): (1 - tan^2(x))/(1 + tan^2(x)) = (1 - sin^2(x)/cos^2(x))/(1 + sin^2(x)/cos^2(x)) Now let's find a common denominator for both the numerator and denominator: = [(cos^2(x) - sin^2(x))/(cos^2(x))] / [(cos^2(x) + sin^2(x))/(cos^2(x))] Next, we can simplify by dividing the numerators and denominators: = (cos^2(x) - sin^2(x))/(cos^2(x) + sin^2(x)) Now we use the Pythagorean identity, which states that sin^2(x) + cos^2(x) = 1, = (cos^2(x) - sin^2(x))/1 Now let's replace 1 in the denominator with 2*cos^2(x) - sin^2(x) + sin^2(x): = (cos^2(x) - sin^2(x))/(2*cos^2(x) - sin^2(x) + sin^2(x)) Next, we can simply write the denominator as 2*cos^2(x): = (cos^2(x) - sin^2(x))/(2*cos^2(x)) Using the double angle identity for cosine which we had at the beginning, we replace cos(2x) = 1 - 2*sin^2(x) with the expression we just derived: cos(2x) = (cos^2(x) - sin^2(x))/(2*cos^2(x)) Hence, we've shown that cos(2x) = (1 - tan^2(x))/(1 + tan^2(x)).