Question

Prove the identity. (1-tanx)^(2)=sec^(2)x-2tanx

Answer 1

( identity has been verified)

Explanation:

To prove the identity (1 - tan(x))^2 = sec(x)^2 - 2tan(x), we can start with the left-hand side:

(1 - tan(x))^2

Expanding this using the square of a binomial formula, we get:

1 - 2tan(x) + tan(x)^2

Next, we can use the identity tan(x)^2 + 1 = sec(x)^2, which follows from the Pythagorean identity for tangent and secant, to substitute for tan(x)^2:

1 - 2tan(x) + (sec(x)^2 - 1)

Simplifying, we get:

sec(x)^2 - 2tan(x)

Therefore, we have shown that the left-hand side (1 - tan(x))^2 is equal to the right-hand side sec(x)^2 - 2tan(x), which proves the identity.