SSC CGL 2025 Tier-I · 19 September 2025, Shift 3 · Quantitative Aptitude · Question 15 of 100

We transcribed this text from the pictures on a candidate's response sheet for this shift; the answer is the key SSC colour-marked on that sheet. Solution last reviewed 11 September 2026.

Where this paper and its answers come from

SSC does not publish CGL question papers or a public answer key. From 2025 the candidate response sheet prints every question and option as a picture, so the text here is transcribed from those pictures for this shift and checked against them independently; where the layout carries meaning, the picture is shown as well. Each answer is the key SSC colour-marked on that sheet. SSC released a tentative key for challenge and later its final one, both only to candidates behind a login, and the sheet does not say which of the two it carries. The solutions are written by UnlockIAS.

Ifsin x + cos x = √2, what is the value of sin x − cos x?

(a)0
(b)1
(c)2
(d)3

Answer

Why

Correct — A. Square the given sum and use sin²x + cos²x = 1.

Square: (sin x + cos x)² = (√2)² = 2
Expand: 1 + 2 sin x cos x = 2, so 2 sin x cos x = 1

Square the difference: (sin x − cos x)² = 1 − 2 sin x cos x
Substitute: 1 − 1 = 0
So sin x − cos x = 0 → option (a)

Why the others are wrong

(b)
1 — 1 needs (sin x − cos x)² = 1, so 2 sin x cos x = 0. Then (sin x + cos x)² would be 1, not the 2 the stem gives.
(c)
2 — 2 is impossible for any x: sin x − cos x = √2 sin(x − 45°), so it never exceeds √2 ≈ 1.41.
(d)
3 — 3 is above the largest value sin x − cos x can take, which is √2 ≈ 1.41. No angle reaches it, whatever the given sum.

Concept

The sum and the difference of sin x and cos x are tied by one identity:

(sin x + cos x)² + (sin x − cos x)² = 2

Given one, the other follows. Here the sum squared is 2, so the difference squared is 0.

√2 is also the largest value sin x + cos x can take. Between 0° and 360° it is reached only at x = 45°, where sin x = cos x.

Check by substitution: at x = 45°, sin x + cos x = √2⁄2 + √2⁄2 = √2, and sin x − cos x = 0.

Key facts

(sin x + cos x)² = 1 + 2 sin x cos x.
(sin x − cos x)² = 1 − 2 sin x cos x.
sin x + cos x and sin x − cos x both lie between −√2 and √2.

Study next

Finding sin x cos x, or sin⁴x + cos⁴x, from a given sin x + cos xMaximum and minimum of a sin x + b cos x, which are ±√(a² + b²)tan x + cot x = 1 ⁄ (sin x cos x)

Common traps

Squaring sin x + cos x as sin²x + cos²x and losing the cross term 2 sin x cos x
Missing that √2 is the maximum of sin x + cos x, which fixes x = 45° at once

The same squaring step drives 23 Sep 2024, 12:30, Quant Q.5, which gives sin A − cos A = (√3 − √2)⁄2 and asks for sin A × cos A.

Related PYQs

No directly related past PYQ was found.