Hey guys can you prove this identity for me and please tell me how you do it. Since its my first time proving identites, I have no idea what to do.
Sin theta + Sin 2 theta
________________________ = Tan theta
1 + Cos theta + Cos 2 theta
Thanks
Hey guys can you prove this identity for me and please tell me how you do it. Since its my first time proving identites, I have no idea what to do.
Sin theta + Sin 2 theta
________________________ = Tan theta
1 + Cos theta + Cos 2 theta
Thanks
$\displaystyle \color{blue}sin(2\theta) = 2sin(\theta)cos(\theta) $
$\displaystyle Numerator= sin(\theta)+2sin(\theta)cos(\theta) = sin(\theta)(1+2cos(\theta))$
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$\displaystyle \color{blue}cos(2\theta) = cos^2(\theta)-sin^2(\theta) $
$\displaystyle \color{blue}sin^2(\theta)+cos^2(\theta)=1$
so $\displaystyle cos(2\theta) = 2cos^2(\theta) -1 $
--------------------------------------
$\displaystyle Denominator ={\color{red} 1}+cos(\theta) + 2cos^2(\theta) {\color{red}-1} = cos(\theta)(1+2cos(\theta))
$
Fraction now becomes
$\displaystyle L.H.S=\frac{sin(\theta){\color{red}(1+2cos(\theta) )}}{cos(\theta){\color{red}(1+2cos(\theta))}} $
$\displaystyle =\frac{sin(\theta)}{cos(\theta)}$
$\displaystyle =tan(\theta)=R.H.S
$
___________________
things in blue are formulas
things in red get canceled