Question

if A=30° prove that : 1+ sin 2 A+cos 2 A/sin A + cos A =2cos A​

Answer 1

Step-by-step explanation:

R.H.S

2cos30° =2×√3/2=√3

L.H.S

$\: \: \frac{1 + \sin(2a) + \cos(2a) }{ \sin(a) + \cos(a) } \\ = > \frac{1 + \sin(2 \times 30) + \cos(2 \times 30) }{ \sin(30) + \cos(30) } \\ = > \frac{1 + \sin(60) + \cos(60) }{ \sin(30) + \cos(30) } \\ = > \frac{1 + \frac{ \sqrt{3} }{2} + \frac{1}{2} }{ \frac{1}{2} + \frac{ \sqrt{3} }{2} } \\ = > \frac{ \frac{2 + \sqrt{3} + 1}{2} }{ \frac{1 + \sqrt{3} }{2} } \\ = > \frac{3 + \sqrt{3} }{2} \times \frac{2}{1 + \sqrt{3} } \\ = > \frac{3 + \sqrt{3} }{1 + \sqrt{3} } \\ = > \frac{3 + \sqrt{3} }{ \sqrt{3 } + 1} \times \frac{ \sqrt{3} - 1}{ \sqrt{3} - 1 } \\ = > \frac{3( \sqrt{3} - 1) + \sqrt{3} ( \sqrt{3} - 1 }{ { \sqrt{3} \: }^{2} - 1} \\ = > \frac{3 \sqrt{3} - 3 + 3 - \sqrt{3} }{3 - 1} \\ = > \frac{3 \sqrt{3} - \sqrt{3} - 3 + 3 }{2} \\ = > \frac{2 \sqrt{3} }{2} \\ = > \sqrt{3}$

Hence

Proved.