Prove that - cos pi/12 - sin pi/12 = 1/root 2 - Maths - Trigonometric Functions

Question

Prove that -
cos pi/12 - sin pi/12 = 1/root 2

Ankush Jain · Accepted Answer

$To Prove: \cos \frac{π}{12} - \sin \frac{π}{12} = \frac{1}{\sqrt{2}}$

Proof :

$LHS = \cos \frac{π}{12} - \sin \frac{π}{12} = \cos 15 ° - \sin 15 ° = \cos (45 ° - 30 °) - \sin (45 ° - 30 °) = \cos 45 ° \cos 30 ° + \sin 45 ° \sin 30 ° - \sin 45 ° \cos 30 ° + \cos 45 ° \sin 30 °$

$= \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} + \frac{1}{\sqrt{2}} \cdot \frac{1}{2} - \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} + \frac{1}{\sqrt{2}} \cdot \frac{1}{2} = \frac{1}{2 \sqrt{2}} + \frac{1}{2 \sqrt{2}} = \frac{2}{2 \sqrt{2}} = \frac{1}{\sqrt{2}}$

= RHS [Hence Proved]