Question

Please help me !! Demonstrate the identity

Answer 1

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

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

$cos(2x) = 1 - 2sin^{2}(x) = 2cos^{2}(x) - 1$
$sin(2x) = 2sinxcosx$

$LHS = \frac{1 - 1 + 2sin^{2}(x) + 2sinxcosx}{1 + 2cos^{2}(x) - 1 + 2sinxcosx}$
$= \frac{2sinx(sinx + cosx)}{2cosx(cosx + sinx)}$
$= \frac{2sinx}{2cosx}$
$= tanx = RHS$

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