Introduction / Context:
This problem explores special transformations of trigonometric functions. The expressions x = cosec θ − sin θ and y = sec θ − cos θ often appear in algebraic manipulation questions where the aim is to derive a compact identity relating x and y. The task is to work out an exact relation involving x^2 and y^2, then match it with one of the given options.
Given Data / Assumptions:
- x = cosec θ − sin θ.
- y = sec θ − cos θ.
- θ is acute, so sin θ and cos θ are positive.
- We must find which of the given algebraic relations between x and y is identically true.
Concept / Approach:
The strategy is to rewrite x and y in terms of sin θ and cos θ only, remove reciprocal functions, and then compute x^2, y^2, and x^2 y^2. Known algebraic simplifications show that such expressions often collapse nicely due to the identity sin^2 θ + cos^2 θ = 1. After simplification we look for a compact expression that equals 1, and match that with the most suitable option.
Step-by-Step Solution:
1. Start with x = cosec θ − sin θ = 1 / sin θ − sin θ.
2. Combine terms: x = (1 − sin^2 θ) / sin θ = cos^2 θ / sin θ.
3. Similarly, y = sec θ − cos θ = 1 / cos θ − cos θ.
4. Combine y: y = (1 − cos^2 θ) / cos θ = sin^2 θ / cos θ.
5. Compute x^2: x^2 = cos^4 θ / sin^2 θ.
6. Compute y^2: y^2 = sin^4 θ / cos^2 θ.
7. Find x^2 y^2: x^2 y^2 = (cos^4 θ / sin^2 θ) * (sin^4 θ / cos^2 θ) = sin^2 θ cos^2 θ.
8. Now compute x^2 + y^2: x^2 + y^2 = cos^4 θ / sin^2 θ + sin^4 θ / cos^2 θ.
9. Bring to a common denominator sin^2 θ cos^2 θ to combine, but instead of fully expanding we evaluate x^2 y^2 (x^2 + y^2 + 3).
10. Substitute x^2 y^2 = sin^2 θ cos^2 θ into x^2 y^2 (x^2 + y^2 + 3).
11. Compute x^2 + y^2 in terms of sin θ and cos θ and simplify; after algebraic manipulation it reduces so that x^2 y^2 (x^2 + y^2 + 3) equals 1.
12. Therefore the identity that always holds is x^2 y^2 (x^2 + y^2 + 3) = 1.
Verification / Alternative check:
We can verify numerically with a convenient acute angle, for example θ = 45 degrees. Then sin θ = cos θ = √2 / 2. This gives x = cosec θ − sin θ = (√2) − (√2 / 2) = √2 / 2, and y = sec θ − cos θ = √2 − √2 / 2 = √2 / 2, so x = y. Then x^2 = y^2 = 1 / 2, and x^2 y^2 = 1 / 4. Plugging into x^2 y^2 (x^2 + y^2 + 3) gives (1 / 4) * (1 / 2 + 1 / 2 + 3) = (1 / 4) * 4 = 1, which confirms option 2 is correct.
Why Other Options Are Wrong:
Option 1: x^2 + y^2 + 3 = 1 would give x^2 + y^2 = −2, which is impossible for real x and y since squares are non negative.
Option 3: x^2 (x^2 + y^2 − 5) = 1 is not satisfied for θ = 45 degrees because this would give (1 / 2) * (1 − 5) = −2, which is not equal to 1.
Option 4: y^2 (x^2 + y^2 − 5) = 1 fails by the same reasoning as option 3 since x^2 = y^2 in the test case.
Common Pitfalls:
Many learners forget to square both the numerator and denominator when finding x^2 and y^2. Another pitfall is attempting to simplify x^2 + y^2 directly without considering a more guided expression such as x^2 y^2 (x^2 + y^2 + 3). It is also easy to mismanage signs when expanding powers of sine and cosine. Careful substitution and patient algebra avoid inconsistencies.
Final Answer:
The correct relation between x and y is
x^2 y^2 (x^2 + y^2 + 3) = 1.
Discussion & Comments