In trigonometry, the real angle X (with 0° ≤ X ≤ 90°) satisfies the equation sin X − cos X = 1. Using this information, determine the exact value of the sum sin X + cos X.

Introduction / Context:
This question tests basic trigonometric identities and reasoning about angles in the first quadrant. Instead of demanding memorisation of special angle values directly, it uses a simple equation relating sin X and cos X to guide you to a specific angle. Once that angle is known, another simple trigonometric expression has to be evaluated. Such questions check both conceptual understanding and careful algebraic manipulation.

Given Data / Assumptions:
- X is an angle measured in degrees with 0° ≤ X ≤ 90° (first quadrant).
- The equation sin X − cos X = 1 holds exactly.
- We must find sin X + cos X for the same angle X.
- Standard trigonometric values for special angles are valid, and sin X and cos X are non negative in this interval.

Concept / Approach:
The strategy is to use the Pythagoras identity sin^2 X + cos^2 X = 1 along with the given relation sin X − cos X = 1. Squaring the given equation helps eliminate the square root behaviour and relate sin X cos X to the identity. Once possible values for sin X and cos X are found, we can quickly compute their sum. Because we are in the first quadrant, we can discard extraneous possibilities based on sign.

Step-by-Step Solution:
1) Start with the equation sin X − cos X = 1. 2) Square both sides: (sin X − cos X)^2 = 1^2. 3) Expand the left side: sin^2 X + cos^2 X − 2 sin X cos X = 1. 4) Use sin^2 X + cos^2 X = 1 to simplify, giving 1 − 2 sin X cos X = 1. 5) Subtract 1 from both sides: −2 sin X cos X = 0, so sin X cos X = 0. 6) Thus either sin X = 0 or cos X = 0. In 0° to 90°, sin X = 0 only at X = 0°, which gives 0 − 1 = −1, not 1. So X must be 90°. 7) At X = 90°, sin 90° = 1 and cos 90° = 0, so sin X + cos X = 1 + 0 = 1.

Verification / Alternative check:
We can directly test the candidate angle X = 90° in the original equation. Substituting gives sin 90° − cos 90° = 1 − 0 = 1, which matches the condition exactly. No other angle in the first quadrant yields sin X − cos X = 1, so this solution is unique. Therefore, the corresponding sum sin X + cos X must be 1, confirming our result.

Why Other Options Are Wrong:
Option B (0) would correspond to sin X = −cos X, which cannot hold in the first quadrant where both are non negative. Option C (1/2) and Option D (2) are typical guesses without solving the equation. Option E (√2) comes from confusing this with the known value of sin 45° + cos 45°, but X is not 45° here. Only Option A matches the exact computed sum from the valid angle X = 90°.

Common Pitfalls:
Learners may try to guess the angle by memory rather than using algebra, which can be risky. Another frequent error is to forget that squaring an equation can introduce extraneous solutions, so the result must always be checked in the original equation. Some students also wrongly assume that sin X and cos X are always equal in the first quadrant, which is only true at 45°. Careful use of identities and verification avoids these mistakes.

Final Answer:
The only first quadrant angle that satisfies sin X − cos X = 1 is X = 90°, and for that angle sin X + cos X = 1.