Question

Perform the transformation.Write cos x in terms of tan x.

Answer 1

`\cos (x)=\sqrt[]{(1)/(1+\tan^2(x))}`

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:

`\begin{gathered} \cos ^2(x)+\sin ^2(x)=1 \\ (\cos ^2(x)+\sin ^2(x))/(\cos ^2(x))=(1)/(\cos ^2(x)) \\ 1+(\sin^2(x))/(\cos^2(x))=(1)/(\cos ^2(x)) \\ 1+\tan ^2(x)=(1)/(\cos^2(x)) \\ (1+\tan ^2(x)).\cos ^2(x)=1 \\ ((1+\tan ^2(x)).\cos ^2(x))/((1+\tan ^2(x)))=(1)/((1+\tan ^2(x)) \\ \cos ^2(x)=(1)/(1+\tan ^2(x)) \\ \sqrt[]{\cos ^2(x)}=\sqrt[]{(1)/(1+\tan^2(x))} \\ \cos (x)=\sqrt[]{(1)/(1+\tan^2(x))} \end{gathered}`

Thus, this is the answer.