# Prove that (1+cotx)/cscx-secx/(tanx+cotx)=cosx?

Jan 23, 2017

#### Explanation:

$\frac{1 + \cot x}{\csc} x - \sec \frac{x}{\tan x + \cot x}$

= $\frac{1 + \cos \frac{x}{\sin} x}{\frac{1}{\sin} x} - \frac{\frac{1}{\cos} x}{\sin \frac{x}{\cos} x + \cos \frac{x}{\sin} x}$

= (sinx+cosx)-(1/cosx)/((sin^2x+cos^2x)/(sinxcosx)

mutiplying numerator and denominator in first term by $\sin x$

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

= $\left(\sin x + \cos x\right) - \frac{1}{\cos} x \times \sin x \cos x$

= $\sin x + \cos x - \sin x$

= $\cos x$