How do you prove (sinx+cosx)(Tanx+cotx)=secx+cscx?

Mar 22, 2018

We have:

$\left(\sin x + \cos x\right) \left(\sin \frac{x}{\cos} x + \cos \frac{x}{\sin} x\right) = \sec x + \csc x$

$\left(\sin x + \cos x\right) \left(\frac{{\sin}^{2} x + {\cos}^{2} x}{\sin x \cos x}\right) = \sec x + \csc x$

$\frac{\sin x + \cos x}{\sin x \cos x} = \sec x + \csc x$

$\sin \frac{x}{\sin x \cos x} + \cos \frac{x}{\sin x \cos x} = \sec x + \csc x$

$\frac{1}{\cos} x + \frac{1}{\sin} x = \sec x + \csc x$

$\sec x + \csc x = \sec x + \csc x$

$L H S = R H S$

Hopefully this helps!