If a = 45° and b = 15°, what is the value of \({\cos (a - b ) - \cos (a + b)} \over {\cos(a - b) + \cos(a + b)}\)?

Question

Question

This question was previously asked in

SSC CGL 2022 Tier-I Official Paper (Held On : 13 Dec 2022 Shift 2)

Answer 1

Given:

a = 45° and b = 15

Concept used:

a² - b² = (a + b)(a - b)

(a - b)² = a² - 2ab + b²

Trigo

Calculation:

(a - b) = 45° - 15° = 30°

(a + b) = 45° + 15° = 60°

Now, \({\cos (a - b ) - \cos (a + b)} \over {\cos(a - b) + \cos(a + b)}\)

⇒ \({\cos 30^\circ - \cos 60^\circ} \over {\cos30^\circ + \cos 60^\circ}\)

⇒ \({\frac {\sqrt3}{2} - \frac {1}{2} } \over {\frac {\sqrt3}{2} + \frac {1}{2} }\)

⇒ \({\sqrt3 - 1} \over {\sqrt3 + 1}\)

⇒ \({(\sqrt3 - 1)(\sqrt3 - 1)} \over {(\sqrt3 + 1)(\sqrt3 - 1)}\)

⇒ \((3 + 1 - 2\sqrt3) \over {3 - 1}\)

⇒ 2 - √3

∴ The simplified value is 2 - √3.

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