1 + cos theta + sin …

LHS {tex} \frac{{1 + \cos \theta + \sin \theta }}{{1 + \cos \theta - \sin \theta }}{/tex}
Dividing numerator and denominator by cos{tex} \theta {/tex}
{tex}= \frac{{\frac{1}{{\cos \theta }} + \frac{{\cos \theta }}{{\cos \theta }} + \frac{{\sin \theta }}{{\cos \theta }}}}{{\frac{1}{{\cos \theta }} + \frac{{\cos \theta }}{{\cos \theta }} - \frac{{\sin \theta }}{{\cos \theta }}}}{/tex}
{tex}= \frac{{\sec \theta + 1 + \tan \theta }}{{\sec \theta + 1 - \tan \theta }}{/tex}
Multiplying and dividing by {tex} \sec \theta + 1 + \tan \theta {/tex}
{tex}= \frac{{\sec \theta + 1 + \tan \theta }}{{\sec \theta + 1 - \tan \theta }} \times \frac{{\sec \theta + 1 + \tan \theta }}{{\sec \theta + 1 + \tan \theta }}{/tex}
{tex}= \frac{{{{(\sec \theta + 1 + \tan \theta )}^2}}}{{{{(\sec \theta + 1)}^2} - {{\tan }^2}\theta }}{/tex}
{tex}= \frac{{{{\sec }^2}\theta + 1 + {{\tan }^2}\theta + 2\sec \theta + 2\tan \theta + 2\sec \theta \tan \theta }}{{1 + {{\sec }^2}\theta + 2\sec \theta - {{\tan }^2}\theta }}{/tex}
Now, {tex} 1 + {\tan ^2}\theta = {\sec ^2}\theta {/tex}
{tex}= \frac{{2{{\sec }^2}\theta + 2\sec \theta + 2\tan \theta + 2\sec \theta \tan \theta }}{{2 + 2\sec \theta }}{/tex}
{tex} = \frac{{2\left[ {\sec \theta (\sec \theta + 1) + \tan \theta (1 + \sec \theta } \right]}}{{2(1 + \sec \theta )}}{/tex}
{tex} = \frac{{(\sec + \tan \theta )(\sec \theta + 1)}}{{(1 + \sec \theta )}}{/tex}
{tex} = \sec \theta + \tan \theta = \frac{1}{{\cos \theta }} + \frac{{\sin \theta }}{{\cos \theta }}{/tex}
{tex}= \frac{{1 + \sin \theta }}{{\cos \theta }} = RHS{/tex}