Question

How do you verify `sin x + cos x + tan x sin x = sec x + cos x tan x`?

Answer 1

Start with the identity `sin^2(x) + cos^2(x) = 1`
and gradually work towards the given equation

Explanation:

`sin^2(x)+cos^2(x) = 1`

divide everything by `cos(x)` ...we'll worry about the case when `color(red)(cos(x) = 0)` later
`color(white)("XXXX")` `color(blue)(sin^2(x)/cos(x) + cos(x) = 1/cos(x))`

since `tan(x) = sin(x)/cos(x)` and `sec(x) = 1/cos(x)`

`color(white)("XXXX")` `color(blue)(sin(x)tan(x) + cos(x) = sec(x))`

rearrange to look closer to our target equation:

`color(white)("XXXX")` `color(blue)(cos(x) + tan(x)sin(x) = sec(x))`

looks like we need to add `sin(x)` to make the left side look like the target

`color(white)("XXXX")` `color(blue)(sin(x) +cos(x)+tan(x)sin(x) = sec(x) + sin(x))`

since `tan(x) = sin(x)/cos(x)`
therefore #sin(x) = cos(x)tan(x)

`color(white)("XXXX")` `color(blue)(sin(x)+cos(x)+tan(x)sin(x) = sec(x) + cos(x)tan(x))`

We have now proven the target equation except for the case `color(red)(cos(x) = 0)`

If `cos(x) = 0` then `tan(x)` is undefined
and since `tan(x)` appears on both sides
`color(white)("XXXX")`we will pretend that `"undefined" = "undefined"`
`color(white)("XXXX")`is adequate to cover this case.