If sin 23° = \(\frac{a}{b}\), then the value of sec 23° - sin 67° is __________.

Question

Question

This question was previously asked in

SSC CGL 2023 Tier-I Official Paper (Held On: 14 Jul 2023 Shift 3)

Answer 1

Given:

sin 23° = a/b

Concepts used:

In a right-angled triangle, applying Pythagoras Theorem,

H² = P² + B²

⇒ P² = H² – B²

P → Perpendicular, B → base, H → Hypotenuse of right-angled triangle

sinθ = P/H

cosθ = B/H

tanθ = P/B

Calculation:

sin23° = P/H = a/b

On comparing,

⇒ P = a units and H = b units

Using Pythagoras' Theorem,

⇒ B² = H² – P²

⇒ B² = b² – a²

⇒ B = √b2 – a2

⇒ sec23° = H/B = b/√b2 – a2

⇒ Cos23° = sin67° = √b2 – a2 /b

sec 23° - sin 67° ⇒ b/√b2 – a² - √b2 – a2 /b

sec 23° - sin 67° ⇒ \( \frac{a^2}{b \sqrt{b^2-a^2}}\)

The value is \( \frac{a^2}{b \sqrt{b^2-a^2}}\)

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