1) Let's begin sketching out a triangle so that we gradually visualize the process and also, we'll make use of some trigonometric identities to help us.
2) Therefore, we can sketch out:
So, in this sketch, we've got the principle. But we need more, we need to make use of a Pythagorean Identity:
[tex]\begin{gathered} \cos ^2(x)+\sin ^2(x)=1 \\ \frac{\cos ^2(x)+\sin ^2(x)}{\cos ^2(x)}=\frac{1}{\cos ^2(x)} \\ 1+\frac{\sin^2(x)}{\cos^2(x)}=\frac{1}{\cos ^2(x)} \\ 1+\tan ^2(x)=\frac{1}{\cos^2(x)} \\ (1+\tan ^2(x)).\cos ^2(x)=1 \\ \frac{(1+\tan ^2(x)).\cos ^2(x)}{(1+\tan ^2(x))}=\frac{1}{(1+\tan ^2(x)} \\ \cos ^2(x)=\frac{1}{1+\tan ^2(x)} \\ \sqrt[]{\cos ^2(x)}=\sqrt[]{\frac{1}{1+\tan^2(x)}} \\ \cos (x)=\sqrt[]{\frac{1}{1+\tan^2(x)}} \end{gathered}[/tex]Thus, this is the answer.
