What is the value of

Correct — D. The stem image reads (5⁄√3 − cosec 60°), and the four option images read (a) 1⁄√3, (b) 2⁄√3, (c) 3√3 and (d) √3.

sin 60° = √3⁄2
cosec 60° = 1 ⁄ sin 60° = 2⁄√3

5⁄√3 − 2⁄√3 = 3⁄√3
3⁄√3 = 3√3 ⁄ (√3 × √3) = 3√3⁄3 = √3 → option (d).

Why the others are wrong

• (a)
  1⁄√3 is what you get by 'cancelling' the 3 on top of 3⁄√3 against the 3 inside the root. You cannot cancel across a radical sign — √3 × √3 = 3, so 3⁄√3 rationalises to √3.
• (b)
  2⁄√3 is cosec 60° itself — the quantity being subtracted, not the result. It is a correct first step marked as the final answer.
• (c)
  3√3 is the rationalisation left half-finished. Multiplying 3⁄√3 top and bottom by √3 gives 3√3⁄3, and the 3s must still cancel to leave √3.

Two things are tested at once: the standard value of cosec 60°, and the habit of rationalising a surd before matching an option.

cosec is the reciprocal of sin, so cosec 60° = 1 ÷ (√3⁄2) = 2⁄√3. Both terms then share the denominator √3 and the subtraction is just 5 − 2 = 3 on top.

The last move is where marks go: 3⁄√3 = √3, because multiplying top and bottom by √3 gives 3√3⁄3. A surd left in a denominator will not match any printed option.

Key facts

• sin 60° = √3⁄2, so cosec 60° = 2⁄√3.
• 3⁄√3 = √3, because multiplying top and bottom by √3 gives 3√3⁄3.
• cosec, sec and cot are the reciprocals of sin, cos and tan respectively.

Common traps

• Writing cosec 60° as √3⁄2, which is sin 60°.
• Leaving the answer as 3⁄√3 and hunting for that form among the options.
• Cancelling the numerator 3 against the 3 inside √3.

No directly related past PYQ was found.