Question

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

Answer 1

Explanation:

`\tan( (x)/(2) ) = ( \sin(x) )/(1 + \cos(x) )`

`\tan( (x)/(2) ) = ( √(1 - \cos(x) ) )/( √(1 + \cos(x) ) )`

So we have

`( √(1 - \cos(x) ) )/( √(1 + \cos(x) ) ) = ( \sin(x) )/( √(1 + \cos(x) ) )`

Next, we then rationalize the numerator so we get

`( √(1 - \cos(x) ) )/( √(1 + \cos(x) ) ) ( √(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) } = ( \sin(x) )/(1 + \cos(x) )`

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

`( \sin(x) )/(1 + \cos(x) ) = ( \sin(x) ) )/(1 + \cos(x) )`