Question

I need help with this practice I’m struggling to solve itIt asks *Verify the identity*(I will send you an additional photo with the answer options for the boxes)

Answer 1

`\sin x(1+\tan ^2x)`
`\sin x\sec ^2x`
`\sin x\cdot(1)/(\cos^2x)`
`(\sin x)/(\cos x)\cdot(1)/(\cos x)`

Step-by-step explanation:

Given the below;

`\sin x+\sin x\tan ^2x=\tan x\sec x`

We'll go ahead and manipulate the left-hand side of the equation following the below steps;

Step 1: Factor out sinx;

`\sin x(1+\tan ^2x)=\tan x\sec x`

Step 2:

Recall the below trig identity;

`\sec ^2x=1+\tan ^2x`

Substituting the above, we'll have;

`\sin x\sec ^2x=\tan x\sec x`

Step 3:

Recall that;

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

So we'll have;

`\sin x\cdot(1)/(\cos^2x)=\tan x\sec x`

Step 4:

We can further simplify as seen below;

`\begin{gathered} \sin x\cdot(1)/(\cos x\cdot\cos x)=\tan x\sec x \\ (\sin x)/(\cos x)\cdot(1)/(\cos x)=\tan x\sec x \end{gathered}`

Step 5:

Recall that;

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

So we can rewrite our equation as;

`\tan x\sec x=\tan x\sec x`