SSC CGL 2025 Tier-I · 12 September 2025, Shift 2 · Quantitative Aptitude · Question 20 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 13 September 2026.

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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.

If tan(90° − A) = √3, what is sin A?

(a)1⁄2
(b)√3⁄2
(c)√2⁄2
(d)3⁄4

Answer

Why

Correct — A. Rewrite the co-function in terms of A first.

tan(90° − A) = cot A, so cot A = √3
cosec²A = 1 + cot²A = 1 + 3 = 4
cosec A = 2 (positive root, as all four options are positive)
sin A = 1 ÷ cosec A = 1⁄2 → option (a)

Check: cot 30° = √3 and sin 30° = 1⁄2 ✓

Why the others are wrong

(b)
√3⁄2 — √3⁄2 is sin 60°, the value you get by setting A = 60°. But tan(90° − A) = √3 makes 90° − A = 60°, so A = 30°.
(c)
√2⁄2 — √2⁄2 is sin 45°. A = 45° would need tan(90° − A) = tan 45° = 1, not √3.
(d)
3⁄4 — 3⁄4 is not a standard-angle value. With sin A = 3⁄4, cos A = √7⁄4 and cot A = √7⁄3 ≈ 0.88, far from √3 ≈ 1.73.

Concept

Co-function identities pair each ratio with its partner at the complementary angle: tan(90° − A) = cot A, sin(90° − A) = cos A, sec(90° − A) = cosec A.

So the question is really cot A = √3, the 30° value from the standard table.

Without the table, cosec²A = 1 + cot²A gives the same result: cosec²A = 4, so sin A = 1⁄2.

The question does not say A is acute. cot A = √3 also holds at A = 210°, where sin A = −1⁄2.

Every option is positive, so the acute reading is the one the options allow.

Key facts

tan(90° − A) = cot A and cot(90° − A) = tan A.
cot 30° = √3 and cot 60° = 1⁄√3.
1 + cot²A = cosec²A.
sin 30° = 1⁄2, sin 45° = √2⁄2, sin 60° = √3⁄2.

Study next

Co-function identities for all six ratios at 90° − AStandard-angle values for 0°, 30°, 45°, 60° and 90°Finding every ratio from one given ratio with a right triangle

Common traps

Solving tan A = √3 instead of tan(90° − A) = √3. That sets A = 60° and gives √3⁄2, the sine of the complement.
Swapping sin 30° and sin 60°. Sine rises from 0° to 90°, so the smaller angle has the smaller sine: 1⁄2 against √3⁄2.

The same value, cot A = √3, starts 19 Sep 2025, 16:00, Quant Q.14, where sin A = 1⁄2 and cos A = √3⁄2 make (1 + sin A)(1 + cos A) = (6 + 3√3)⁄4.

Allied-angle conversions, cos(90 − θ) = sin θ and sin(90 + θ) = cos θ, collapse a long fraction to −1 on 17 Sep 2024, 12:30, Quant Q.17.

Related PYQs

No directly related past PYQ was found.

Topic: Trigonometric ratio, Degree and Radian Measures, Standard Identities, Complementary angles, Heights and Distances — all 131 SSC CGL questions on it, every solved shift.