What is the value of tan(π/4 + A) multiplied by tan(3π/4 + A) expressed in simplest form?

Introduction / Context:
This trigonometry question focuses on tangent of compound angles involving π/4 and 3π/4. It tests your understanding of how tangent behaves when the angle is shifted by π/2 or π, and how to use standard tangent addition and subtraction formulas to simplify expressions involving sums of angles.

Given Data / Assumptions:

• Expression: tan(π/4 + A) × tan(3π/4 + A)
• A is a real angle.
• Angles are in radians and tangent is defined for the angles considered (denominators in the formulas are not zero).

Concept / Approach:
We use the identities for tangent of sums and the periodicity properties of tangent. Two important relations are:
tan(π/4 + A) = (1 + tan A)/(1 - tan A)
tan(3π/4 + A) can be related to tan(π/4 - A) by using the identity tan(π - θ) = -tan θ.
Once each tangent is written in terms of tan A, we multiply the expressions and simplify the resulting fraction algebraically.

Step-by-Step Solution:
Step 1: Let t = tan A for simplicity. Step 2: Use the tangent addition formula for tan(π/4 + A): tan(π/4 + A) = (1 + t)/(1 - t). Step 3: Observe that 3π/4 = π - π/4. So 3π/4 + A = π - (π/4 - A). Step 4: Use tan(π - θ) = -tan θ with θ = π/4 - A, giving tan(3π/4 + A) = -tan(π/4 - A). Step 5: Apply the formula tan(π/4 - A) = (1 - t)/(1 + t). Step 6: Therefore tan(3π/4 + A) = - (1 - t)/(1 + t). Step 7: Multiply the two tangents: tan(π/4 + A) × tan(3π/4 + A) = [(1 + t)/(1 - t)] × [ - (1 - t)/(1 + t) ]. Step 8: Cancel the common factors (1 + t) and (1 - t) in numerator and denominator to get the result -1.

Verification / Alternative check:
Choose a simple angle for A, for example A = 0. Then tan(π/4 + 0) = tan(π/4) = 1 and tan(3π/4 + 0) = tan(3π/4) = -1. Their product is 1 × ( -1 ) = -1, which matches the simplified algebraic result. This quick numeric check confirms that the reasoning is correct.

Why Other Options Are Wrong:
1 would be the product if both tangents were equal in magnitude and sign, which is not the case here. 0 would require at least one of the factors to be zero, but tangent at π/4 + A and 3π/4 + A is not zero for a general A. cot(A/2) and tan A involve different trigonometric relationships and do not arise naturally from the product of these two specific tangent expressions.

Common Pitfalls:
Students sometimes confuse tan(π - θ) with tan(π/2 - θ) and apply the wrong identity. Another common error is mishandling the negative sign when using tan(π - θ) = -tan θ. Carefully rewriting 3π/4 + A in the form π - something and substituting correctly prevents these mistakes.

Final Answer:
The value of tan(π/4 + A) × tan(3π/4 + A) is -1.