Question

50 POINTSSSS
Prove It
$\frac{sin \frac{a}{2} + \sin(a) }{1 + \cos( \frac{a}{2} ) + \cos(a) }$
$= \tan( \frac{a}{2} )$
​

Answer 1

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Answer 2

Formulas:

(i) sin(2a) = 2sin(a)cos(a)

(ii) cos(2a) = cos²a - sin²a

(iii) sin²(a) + cos²(a) = 1 same sin²(a/2) + cos²(a/2) = 1

Explanation:

$\sf \rightarrow \dfrac{sin(a)+sin\left(\frac{a}{2}\right)}{1+cos\left(\frac{a}{2}\right)+cos(a)}$

use (i)

$\sf \rightarrow \dfrac{2sin(\frac{a}{2})cos(\frac{a}{2})+sin\left(\frac{a}{2}\right)}{cos\left(\frac{a}{2}\right)+cos(a)+1}$

use (ii)

$\sf \rightarrow \dfrac{2sin(\frac{a}{2})cos(\frac{a}{2})+sin\left(\frac{a}{2}\right)}{cos\left(\frac{a}{2}\right)+cos^2(\frac{a}{2}) - sin^2(\frac{a}{2})+1}$

use (iii)

$\sf \rightarrow \dfrac{2sin(\frac{a}{2})cos(\frac{a}{2})+sin\left(\frac{a}{2}\right)}{cos\left(\frac{a}{2}\right)+cos^2(\frac{a}{2}) - sin^2(\frac{a}{2})+sin^2(\frac{a}{2})+cos^2(\frac{a}{2})}$

simplify

$\sf \rightarrow \dfrac{2sin(\frac{a}{2})cos(\frac{a}{2})+sin\left(\frac{a}{2}\right)}{cos\left(\frac{a}{2}\right)+2cos^2(\frac{a}{2})}$

take common

$\sf \rightarrow \dfrac{sin(\frac{a}{2})(2cos(\frac{a}{2})+1)}{cos\left(\frac{a}{2}\right)(1+2cos(\frac{a}{2}))}$

cancel out

$\sf \rightarrow \dfrac{sin(\frac{a}{2})\left }{cos\left(\frac{a}{2}\right)}$

equals

$\sf \rightarrow tan(\frac{a}{2})$; Hence, proved.