How do I prove a trig identity??

[phil]

how can I prove:

(1+2sinxcosx)/(sinx+cosx)=(sinx + cosx)

 

[royhaas]

Use \(\displaystyle \sin^2(x)+\cos^2(x)=1\).

 

[pka]

\(\displaystyle \begin{array}{l} \left( {\sin (x) + \cos (x)} \right)\frac{{\left( {\sin (x) + \cos (x)} \right)}}{{\left( {\sin (x) + \cos (x)} \right)}} \\ = \frac{{\left( {\sin ^2 (x) + 2\sin (x)\cos (x) + \cos ^2 (x)} \right)}}{{\left( {\sin (x) + \cos (x)} \right)}} \\ = \frac{{\left( {1 + 2\sin (x)\cos (x)} \right)}}{{\left( {\sin (x) + \cos (x)} \right)}} \\ \end{array}\)

 

[phil]

Thanks I never actually saw that. Our teacher always says to start on the most complex side, and so I never even considered the right side of the expression.

 

[Loren]

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

\(\displaystyle \frac{(\sin x + \cos x)(\sin x + \cos x)}{\sin x + \cos x} = \sin x + \cos x\)