Question

Prove or disprove the identity. If you find the identity is true, state the firstline of the proof. If you find the identity is false, write the correct equation byreplacing the right side.Sin x tan x =1 + cos2x _______ Cos x

Answer 1

Given

`\sin x\tan x=(1+\cos 2x)/(\cos x)`

Procedure

Let's develop the right-hand side of the equation

`\begin{gathered} \sin x\tan x=(2\cos ^2x)/(\cos x) \\ \sin x\tan x=2\cos x \\ \sin x\cdot(\sin x)/(\cos x)=2\cos x \\ (\sin^2x)/(\cos x)=2\cos x_{} \end{gathered}`

The identity is false

write the correct equation by replacing the right side.

`\begin{gathered} \sin x\tan x=(\sin x\sin x)/(\cos x) \\ \sin x\tan x=(\sin ^2x)/(\cos x) \\ \sin x\tan x=(1-\cos ^2x)/(\cos x) \\ \sin x\tan x=(1-(1)/(2)-(1)/(2)\cos 2x)/(\cos x) \\ \sin x\tan x=(1-\cos 2x)/(2\cos x) \end{gathered}`