Simplify [1 − sin²32° + 1⁄(1 + tan²58°)].

Correct — C. The stem is an image; it reads Simplify [1 − sin²32° + 1/(1 + tan²58°)].

Take the first part with the Pythagorean identity.
1 − sin²32° = cos²32°

Take the denominator with the second identity.
1 + tan²58° = sec²58°, so 1/(1 + tan²58°) = cos²58°

Now use complementary angles, since 58° = 90° − 32°.
cos 58° = sin 32°, so cos²58° = sin²32°

Add the two parts.
cos²32° + sin²32° = 1 → option (c).

Why the others are wrong

• (a)
  √3 is the value of tan 60° from the standard-angle table and answers nothing here. Every term in this expression is a square, and the squares cancel to a pure number.
• (b)
  1/2 is cos²45° — the value if the angle is misread as 45° and the second term is dropped. Both terms are needed, because together they are sin²32° + cos²32°.
• (d)
  sin32° — sin 32° is a first-power ratio, but no squared identity reduces to one. It also leaves an angle standing in the answer, which is exactly what the identity removes.

Three standard identities do all the work, and each is meant to be recognised on sight.

1 − sin²θ = cos²θ and 1 + tan²θ = sec²θ are the Pythagorean identities. The second is why the reciprocal 1/(1 + tan²θ) is really cos²θ.

cos(90° − θ) = sin θ is the complementary rule, and 32° + 58° = 90° is the signal that SSC planted it. Once 58° becomes 90° − 32°, the two terms are cos²32° and sin²32°, which add to exactly 1.

Key facts

• sin²θ + cos²θ = 1 for every angle θ.
• 1 + tan²θ = sec²θ, so 1/(1 + tan²θ) = cos²θ.
• cos(90° − θ) = sin θ, which makes cos 58° equal to sin 32°.
• 32° and 58° add to 90°, the standing hint that a complementary-angle step is intended.

Common traps

• Reaching for a calculator value of sin 32° instead of using an identity
• Turning 1/(1 + tan²58°) into sin²58° rather than cos²58°
• Stopping at cos²32° + cos²58° without noticing the angles are complementary

No directly related past PYQ was found.