Proof:

Given: tanA = √2 - 1

To prove: tanA/1 - tan^2A = √2/4

Proof by substitution:

Let's start by substituting the given value of tanA into the expression we need to prove:

√2 - 1 / 1 - (√2 - 1)^2

Simplifying the expression:

√2 - 1 / 1 - (√2 - 1)(√2 - 1)

√2 - 1 / 1 - (2 - 2√2 + 1)

√2 - 1 / 1 - 2 + 2√2 - 1

√2 - 1 / -2 + 2√2

Simplifying the denominator:

To simplify the denominator, we can multiply both the numerator and denominator by the conjugate of the denominator:

(√2 - 1)(2 + 2√2) / (-2 + 2√2)(2 + 2√2)

Simplifying the denominator:

(-2√2 + 2 - √2 + 1) / (-4 + 8)

(-√2 + 3) / 4

Simplifying the numerator:

Expanding the numerator:

-2√2 + 2 - √2 + 1

-√2 - √2 + 2 + 1

-2√2 - 1

Final expression:

Substituting the simplified numerator and denominator back into the expression:

(-2√2 - 1) / 4

Now, we need to prove that this expression is equal to √2/4.

Proving the equality:

To prove that (-2√2 - 1) / 4 = √2/4, we can simplify both sides of the equation:

-2√2 - 1 = √2

Squaring both sides of the equation:

(-2√2 - 1)^2 = (√2)^2

4(2) + 1 + 4√2 = 2

8 + 1 + 4√2 = 2

9 + 4√2 = 2

4√2 = -7

This is not a valid equation, so we can conclude that the expression (-2√2 - 1) / 4 is not equal to √2/4.

Therefore, we cannot prove the given expression using the given value of tanA = √2 - 1.