Question

How do you prove `sin(x) x tan(x) + cos(x) = sec(x)`?

Answer 1

I strongly assume that you would like to prove the identity

`sin(x) tan(x) + cos(x) = sec(x)`

without the `x` between `sin(x)` and `tan(x)`.
(Maybe it was a "times" that was confused with "`x`" on a hand-written note).

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We will need the following identities:

[1] `" " tan(x) = sin(x)/cos(x)`

[2] `" " sec(x) = 1/cos(x)`

[3] `" " sin^2(x) + cos^2(x) = 1`

Let's start at the left side and try to get to the right side:

`sin(x) tan(x) + cos(x) stackrel("[1] ")(=) sin(x) * sin(x)/cos(x) + cos(x)`

`= sin^2(x)/cos(x) + cos(x) * color(blue)(cos(x)/cos(x))`

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

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

`stackrel("[3] ")(=) 1 / cos(x)`

`stackrel("[2] ")(=) sec(x)`

q.e.d.