If 1⁄(1 − sin θ) + 1⁄(1 + sin θ) = 4 sec θ, (0 < θ < 90°), then the value of (cot θ + cosec θ) is:

Correct — D. Add the two fractions first — the denominators make a difference of squares.

1⁄(1 − sinθ) + 1⁄(1 + sinθ)
= [(1 + sinθ) + (1 − sinθ)] ⁄ (1 − sin²θ)
= 2 ⁄ cos²θ = 2 sec²θ

Set that against the right-hand side:
2 sec²θ = 4 secθ → secθ = 2 → θ = 60°

cot 60° + cosec 60° = 1⁄√3 + 2⁄√3
= 3⁄√3 = √3 → option (d).

Why the others are wrong

• (a)
  √6 answers to no angle here. The equation pins θ at 60° and at nothing else between 0° and 90°, and there cot θ + cosec θ is √3.
• (b)
  √5 fails the squaring check: (cot θ + cosec θ)² = 1⁄3 + 4⁄3 + 4⁄3 = 3, so the value is √3. Squaring your answer is the fastest way to test a surd.
• (c)
  √2 is sec 45°, where you land if 2 sec²θ = 4 secθ is cancelled as sec²θ = 2 rather than secθ = 2 — one secθ too many taken out.

Two moves finish this. Over the common denominator (1 − sinθ)(1 + sinθ) = 1 − sin²θ = cos²θ, the numerators add to 2, so the left side is 2 sec²θ.

Then 2 sec²θ = 4 secθ is linear in secθ once you divide by secθ, which is safe because secθ is never zero.

That leaves secθ = 2, so cosθ = 1⁄2 and θ = 60°, the single angle strictly between 0° and 90° with that cosine.

Finally cot 60° = 1⁄√3 and cosec 60° = 2⁄√3, and the sum is 3⁄√3 = √3.

Key facts

• (1 − sinθ)(1 + sinθ) = 1 − sin²θ = cos²θ.
• 1⁄(1 − sinθ) + 1⁄(1 + sinθ) simplifies to 2 sec²θ.
• cot θ + cosec θ = (1 + cos θ) ⁄ sin θ, which at θ = 60° gives √3.
• sec 60° = 2, cos 60° = 1⁄2, cot 60° = 1⁄√3 and cosec 60° = 2⁄√3.

Common traps

• Cancelling secθ carelessly and solving sec²θ = 2, which gives θ = 45°
• Stopping at θ = 60° and reporting sec 60° instead of cot θ + cosec θ

No directly related past PYQ was found.