Solution
[tex]\mleft(\tan x+secx\mright)\mleft(1-\sin x\mright)=cosx[/tex][tex]Explanation\colon[/tex][tex]\begin{gathered} \tan x=\frac{\sin x}{\cos x} \\ \sec x=\frac{1}{\cos x} \\ Startthesimplificationprocess\colon \end{gathered}[/tex][tex]\begin{gathered} \mleft(\frac{\sin x}{\cos x}+\frac{1}{\cos x}\mright)(1-\sin x) \\ (\frac{\sin x+1)}{\cos x}(1-\sin x) \end{gathered}[/tex][tex]\begin{gathered} \frac{1-\sin ^2x}{\cos x} \\ \frac{\cos ^2x}{\cos x} \end{gathered}[/tex]Now, rearrange the pythagorean identity
[tex]\begin{gathered} \sin ^2x+\cos ^2x=1 \\ \cos ^2x=1-\sin ^2x \end{gathered}[/tex][tex]\begin{gathered} \frac{(\cos x)(\cos x)}{\cos x} \\ =\cos x \end{gathered}[/tex][tex]\mleft(\tan x+secx\mright)\mleft(1-\sin x\mright)=cosx[/tex]PROOF
