Question

Factor; then use fundamental identities to simplify the expression below and determine which of the following is NOT equivalent tan^3x-tan^2x+tanx-1a.sec^2x(sinx-cosx/cosx)b.sinx-cosx/(cos^3x)c.sec^2x-tan^2xd. (sinx/cos^3x)-sec^2xe.(tanx)(sec^2x)-sec^2x

Answer 1

We have the following expression

`\tan ^3x-\tan ^2x+\tan x-1`

Factoring

`\begin{gathered} \tan ^2x(\tan x-1)+\tan x-1 \\ (\tan x-1)(\tan ^2x+1) \end{gathered}`

Now, we need to compare every option

For option E = Equivalent

`\begin{gathered} (\tan x-1)(\tan ^2x+1) \\ (\tan x-1)\cdot(\sec ^2x) \\ \tan x\cdot\sec ^2x-\sec ^2x \end{gathered}`

For option D = Equivalent

`\begin{gathered} \tan x\cdot\sec ^2x-\sec ^2x \\ (\sin x)/(\cos x)\cdot(1)/(\cos^2x)-\sec ^2x \\ (\sin x)/(\cos^3x)-\sec ^2x \end{gathered}`

For option C = Not Equivalent

`\begin{gathered} \tan x\cdot\sec ^2x-\sec ^2x \\ \text{ we can not derive any expression like option C} \end{gathered}`

For option B = Equivalent

`\begin{gathered} \tan x\cdot\sec ^2x-\sec ^2x \\ (\sin x)/(\cos x)\cdot(1)/(\cos^2x)-(1)/(\cos^2x) \\ (\sin x)/(\cos^3x)-(\cos x)/(\cos^3x) \\ (\sin x-\cos x)/(\cos^3x) \end{gathered}`

For option A = Equivalent

`\begin{gathered} (\sin x-\cos x)/(\cos^3x) \\ (1)/(\cos^2x)\cdot\frac{\sin x-\cos x}{\cos ^{}x} \\ \sec ^2x\cdot\frac{\sin x-\cos x}{\cos ^{}x} \end{gathered}`