SSC CGL 2024 Tier-I · 26 September 2024, Shift 2 · Quantitative Aptitude · Question 14 of 100

Text as printed on a candidate's response sheet for this shift; answer per SSC's tentative key as marked on that sheet. Solution last reviewed 10 September 2026.

Where this paper and its answers come from

SSC does not publish CGL question papers or a public answer key. The questions and options here are reproduced as printed on a candidate's own response sheet for this shift — the Commission's text, unchanged. Each answer is SSC's tentative key as marked on that sheet; SSC allowed challenges to it and never publishes a final key. The solutions are written by UnlockIAS.

If , then is equal to:

Answer

Why

Correct — B. The stem is printed as images: p is defined as sinA ⁄ (1 + cosA), and the quantity wanted is sinA ⁄ (1 − cosA).

Multiply the two together and watch the denominator collapse:
p × sinA ⁄ (1 − cosA) = sin²A ⁄ [(1 + cosA)(1 − cosA)]
(1 + cosA)(1 − cosA) = 1 − cos²A = sin²A
so the product is 1

The second expression is therefore 1 ⁄ p, which is what option (b) shows. Check at A = 60°: p = (√3 ⁄ 2) ⁄ (3 ⁄ 2) = 1 ⁄ √3, and sinA ⁄ (1 − cosA) = (√3 ⁄ 2) ⁄ (1 ⁄ 2) = √3.

Why the others are wrong

(a)
Option (a) shows p − 1. At A = 60°, p ≈ 0.577, so p − 1 ≈ −0.42 — negative, while sinA ⁄ (1 − cosA) is positive for every A strictly between 0° and 180°.
(c)
Option (c) shows 1 ⁄ (1 − p). At A = 60° that is 1 ⁄ 0.42 ≈ 2.37, against the true √3 ≈ 1.73. Subtracting p from 1 has no identity standing behind it.
(d)
Option (d) shows 1 ⁄ (p + 1). At A = 60° it gives about 0.63, smaller than p itself, whereas the wanted value is p's reciprocal and so must exceed 1 whenever p is below 1.

Concept

The engine is 1 − cos²A = sin²A.

Rationalise sinA ⁄ (1 − cosA) by multiplying top and bottom by the conjugate (1 + cosA). The denominator becomes sin²A, one sinA cancels, and what is left is (1 + cosA) ⁄ sinA — exactly the reciprocal of p.

In half-angle language p = tan(A ⁄ 2) and sinA ⁄ (1 − cosA) = cot(A ⁄ 2), and those two are reciprocals by definition. Either reading reaches 1 ⁄ p, so use whichever you recall faster.

Because the expressions live in images, the row's plain text reads only as 'If , then is equal to:'. Nothing is missing from the paper — the algebra is in the pictures, and all four options are pictures too.

Key facts

sinA ⁄ (1 + cosA) = (1 − cosA) ⁄ sinA = tan(A ⁄ 2).
sinA ⁄ (1 − cosA) = (1 + cosA) ⁄ sinA = cot(A ⁄ 2).
The product of those two expressions is 1 wherever both are defined, which is why one is the reciprocal of the other.
Rationalising a trigonometric fraction means multiplying above and below by the conjugate of the denominator.

Study next

Rationalising trigonometric fractions using the conjugate of the denominatorThe half-angle forms tan(A ⁄ 2) and cot(A ⁄ 2)Conjugate pairs such as secθ + tanθ and secθ − tanθ, whose product is 1

Common traps

Cancelling sinA across the two fractions as though 1 + cosA and 1 − cosA were interchangeable.
Multiplying by (1 − cosA) instead of the conjugate, which leaves (1 − cosA)² underneath and gets you nowhere.
Answering with p, when the question asks for the expression that inverts it.

SSC names one expression with a letter and then asks for a partner expression that turns out to be its reciprocal or its conjugate. The same move is set at 25 Sep 2024, 16:00, Quant Q.14, where secθ + tanθ = x and sinθ is wanted.

Related PYQs

No directly related past PYQ was found.