Prove root 3 cosec 20 - sec 20= 4 - Maths - Trigonometric Functions

Question

Prove
root 3 cosec 20 - sec 20= 4

Ankush Jain · Accepted Answer

$To Prove: \sqrt{3} \cos ec 20 ° - \sec 20 ° = 4$

Proof :

$L.H.S. = \frac{\sqrt{3}}{\sin 20 °} - \frac{1}{\cos 20 °} = \frac{\sqrt{3} \cos 20 ° - \sin 20 °}{\sin 20 ° \cos 20 °}$

Multiply and divide L.H.S. by 4, we get

$L.H.S. = 4 [\frac{\frac{\sqrt{3}}{2} \cos 20 ° - \frac{1}{2} \sin 20 °}{2 \sin 20 ° \cos 20 °}] = 4 [\frac{\cos 30 ° \cdot \cos 20 ° - \sin 30 ° \cdot \sin 20 °}{\sin (20 ° + 20 °) + \sin (20 ° - 20 °)}] = 4 [\frac{\cos (30 ° + 20 °)}{\sin 40 °}]$

$= 4 (\frac{\cos 50 °}{\sin 40 °}) = 4 [\frac{\cos (90 ° - 40 °)}{\sin 40 °}] = 4 (\frac{\sin 40 °}{\sin 40 °}) = 4$

= R.H.S. [Hence Proved]