If tan(x)⁄sec(x) = 1⁄2, then the value of (sin(x) + cos(x))² is ____.

Correct — B. The ratio on the left collapses to a single function.

tan x / sec x = (sin x / cos x) × cos x = sin x
So sin x = 1/2, and for an acute x, cos x = √3/2

Expand the square using sin²x + cos²x = 1:
(sin x + cos x)² = 1 + 2 sin x cos x
= 1 + 2 × (1/2) × (√3/2) = 1 + √3/2

1 + √3/2 = 2/2 + √3/2 = (2 + √3)/2 → option (b)

Why the others are wrong

• (a)
  (2 − √3)/3 gets both halves wrong. The denominator has to be 2, because 1 + √3/2 is written over 2, and the sign has to be plus, because sin x and cos x are both positive for an acute x.
• (c)
  (2 − √3)/2 is 1 − √3/2, the value when cos x = −√3/2. sin x = 1/2 does also allow x = 150°, but with no quadrant stated the reading taken here is the acute angle, where cos x is positive.
• (d)
  (2 + √3)/3 puts the right numerator over the wrong denominator. 1 + √3/2 = 2/2 + √3/2, so the common denominator is 2 and never 3.

Two moves do the whole question.

The first is seeing that tan x / sec x = sin x. Dividing by sec x is the same as multiplying by cos x, and (sin x / cos x) × cos x leaves sin x, so the ratio is a disguise for a simple value.

The second is the expansion (sin x + cos x)² = 1 + 2 sin x cos x, which follows from sin²x + cos²x = 1. Once sin x and cos x are numbers, nothing is left but arithmetic.

Key facts

• tan x / sec x simplifies to sin x, and cot x / cosec x simplifies to cos x.
• (sin x + cos x)² = 1 + 2 sin x cos x, and (sin x − cos x)² = 1 − 2 sin x cos x.
• sin x = 1/2 gives x = 30° in the first quadrant, where cos x = √3/2.

Common traps

• Cancelling tan and sec the wrong way round and reaching cos x instead of sin x
• Dropping the 2 sin x cos x cross term when squaring a sum
• Writing 1 + √3/2 as (2 + √3)/3

No directly related past PYQ was found.