Find the value of the following expression.

Correct — B. Multiply the fraction under the root, top and bottom, by (1 + sin θ).

(1 + sin θ) ⁄ (1 − sin θ) = (1 + sin θ)² ⁄ (1 − sin²θ)
1 − sin²θ = cos²θ
So the root is √[(1 + sin θ)² ⁄ cos²θ] = (1 + sin θ) ⁄ cos θ
Split it: 1 ⁄ cos θ + sin θ ⁄ cos θ = sec θ + tan θ → option (b)

Check at θ = 30°: (1 + 0.5) ⁄ (1 − 0.5) = 3 and √3 ≈ 1.732, while sec 30° + tan 30° ≈ 1.155 + 0.577 = 1.732.

Why the others are wrong

• (a)
  cosec θ + cot θ is (1 + cos θ) ⁄ sin θ, the value of the cosine twin √[(1 + cos θ) ⁄ (1 − cos θ)]. At 30° it is about 3.73, not 1.73.
• (c)
  cosec θ + tan θ mixes the two families: 1 ⁄ sin θ and sin θ ⁄ cos θ share no denominator and never combine into this root. At 30° it is about 2.58.
• (d)
  sec θ + cot θ keeps the right first term and spoils the second. Splitting (1 + sin θ) ⁄ cos θ gives sin θ ⁄ cos θ, which is tan θ. At 30° it is about 2.89.

The whole item is one manoeuvre: multiply by the conjugate. (1 − sin θ)(1 + sin θ) is 1 − sin²θ, which is cos²θ, and a square root over a perfect square disappears.

What is left, (1 + sin θ) ⁄ cos θ, splits into sec θ + tan θ. The pair is worth memorising, because the same shape returns as √[(1 − sin θ) ⁄ (1 + sin θ)] = sec θ − tan θ.

The cosine version behaves identically and produces cosec θ + cot θ, which is why that appears among the options.

Key facts

• √[(1 + sin θ) ⁄ (1 − sin θ)] = sec θ + tan θ.
• √[(1 − sin θ) ⁄ (1 + sin θ)] = sec θ − tan θ, and the two expressions multiply to sec²θ − tan²θ = 1.
• (1 − sin θ)(1 + sin θ) = 1 − sin²θ = cos²θ.

Common traps

• Multiplying by (1 − sin θ) instead of (1 + sin θ), which leaves the root no simpler.
• Splitting (1 + sin θ) ⁄ cos θ into sec θ + sin θ and losing the cos θ under the second term.
• Answering with the cosec-cot pair, which is the cosine version of the same identity.

No directly related past PYQ was found.