Question

Prove that: 1-(tan$)^2\1+(tan$)^2= ( cos$+sin$)(cos$-sin$)

Answer 1

Interpret the statement as
(1 − tan^2 x) / (1 + tan^2 x) = (cos x + sin x)(cos x − sin x).

Proof. For x with cos x ≠ 0 (so tan x is defined),
write tan x = sin x / cos x and multiply numerator and denominator by 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))
= ((cos^2 x − sin^2 x)/cos^2 x)/((cos^2 x + sin^2 x)/cos^2 x)
= (cos^2 x − sin^2 x)/(cos^2 x + sin^2 x).

Using cos^2 x + sin^2 x = 1, this equals cos^2 x − sin^2 x, which factors as
(cos x + sin x)(cos x − sin x).

Thus (1 − tan^2 x)/(1 + tan^2 x) = (cos x + sin x)(cos x − sin x) (for all x where tan x is defined).