Question

1. Prove the following trig identity
tan(x/2)=sinx/1+cosx​

Answer 1

Step-by-step explanation:

$\tan( \frac{x}{2} ) = \frac{ \sin(x) }{1 + \cos(x) }$

$\tan( \frac{x}{2} ) = \frac{ \sqrt{1 - \cos(x) } }{ \sqrt{1 + \cos(x) } }$

So we have

$\frac{ \sqrt{1 - \cos(x) } }{ \sqrt{1 + \cos(x) } } = \frac{ \sin(x) }{ \sqrt{1 + \cos(x) } }$

Next, we then rationalize the numerator so we get

$\frac{ \sqrt{1 - \cos(x) } }{ \sqrt{1 + \cos(x) } } \frac{ \sqrt{1 + \cos(x) } }{1 + \cos(x) } = \frac{ \sin(x) }{ {1 + \cos(x) } }$

The denominator should be square root so to let you know .

So know we ahev

$\frac{ \sqrt{1 - \cos {}^{2} (x) } }{1 + \cos(x) } = \frac{ \sin(x) }{1 + \cos(x) }$

$\frac{ \sqrt{ \sin {}^{2} (x) } }{1 + \cos(x) } = \frac{ \sin(x) }{ {1 + \cos(x) } }$

$\frac{ \sin(x) }{1 + \cos(x) } = \frac{ \sin(x) ) }{1 + \cos(x) }$