If (sin θ + cos θ) / (sin θ − cos θ) = 3 for an angle θ where sin θ ≠ cos θ, then what is the value of the expression sin^4 θ − cos^4 θ?

Introduction / Context:
This trigonometry question involves manipulating an equation with a ratio of sums and differences of sine and cosine. The goal is to evaluate a higher power expression sin^4 θ − cos^4 θ. Such questions test comfort with identities, algebraic manipulation, and recognizing patterns like difference of squares.

Given Data / Assumptions:

(sin θ + cos θ) / (sin θ − cos θ) = 3.
sin θ ≠ cos θ, so the denominator is non zero.
We need to find sin^4 θ − cos^4 θ.

Concept / Approach:
The expression sin^4 θ − cos^4 θ can be written as (sin^2 θ − cos^2 θ)(sin^2 θ + cos^2 θ). Since sin^2 θ + cos^2 θ = 1, the expression simplifies to sin^2 θ − cos^2 θ. Also, sin^2 θ − cos^2 θ equals (sin θ + cos θ)(sin θ − cos θ). Therefore, once we know the product (sin θ + cos θ)(sin θ − cos θ), we directly obtain sin^4 θ − cos^4 θ. Given their ratio, we can express the squares of the sum and difference in terms of sin θ cos θ and solve for that product.

Step-by-Step Solution:
Let S = sin θ + cos θ and D = sin θ − cos θ.Given S / D = 3, so S = 3D.Compute S^2: S^2 = (sin θ + cos θ)^2 = sin^2 θ + 2 sin θ cos θ + cos^2 θ = 1 + 2k, where k = sin θ cos θ.Compute D^2: D^2 = (sin θ − cos θ)^2 = sin^2 θ − 2 sin θ cos θ + cos^2 θ = 1 − 2k.From S = 3D we get S^2 = 9D^2.So 1 + 2k = 9(1 − 2k) = 9 − 18k.Rearrange: 1 + 2k = 9 − 18k gives 20k = 8, so k = 8 / 20 = 2 / 5.Now D^2 = 1 − 2k = 1 − 4/5 = 1/5.Then S * D = (sin θ + cos θ)(sin θ − cos θ) = sin^2 θ − cos^2 θ = D^2 * (S / D) = D * S = (±√(1/5)) * (±3√(1/5)) = 3/5.Thus sin^2 θ − cos^2 θ = 3/5.Finally, sin^4 θ − cos^4 θ = (sin^2 θ − cos^2 θ)(1) = 3/5.

Verification / Alternative check:
Once we know sin^2 θ − cos^2 θ = 3/5 and sin^2 θ + cos^2 θ = 1, we can solve for sin^2 θ and cos^2 θ explicitly and check the original ratio. Solving the system gives sin^2 θ = 4/5 and cos^2 θ = 1/5 (or vice versa, depending on signs). Suitable choices of signs for sin θ and cos θ can then be used to verify that (sin θ + cos θ) / (sin θ − cos θ) equals 3. This confirms that sin^4 θ − cos^4 θ = 3/5 is consistent with the given condition.

Why Other Options Are Wrong:
The values 4/3, 3/4, 5/3, and 1/3 do not match the product (sin θ + cos θ)(sin θ − cos θ) computed from the ratio S / D = 3. They arise from incorrect manipulation of the squares or from forgetting that sin^4 θ − cos^4 θ reduces to sin^2 θ − cos^2 θ. Only 3/5 respects both the identity and the given ratio.

Common Pitfalls:
Some students attempt to square the ratio directly without tracking the relationship between S^2 and D^2. Others incorrectly treat sin^4 θ − cos^4 θ as (sin θ − cos θ)^4, which is not true. Remember to use the difference of squares factorization, and be careful when introducing the variable k for sin θ cos θ. Systematic algebra prevents such mistakes.

Final Answer:
The value of sin^4 θ − cos^4 θ is 3/5.