Question

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

Answer 1

Recall the following quotient, Pythagorean, and reciprocal identities:

`1. color(red)(tanx=sinx/cosx)`

`2. color(darkorange)(sin^2x+cos^2x=1)`

`3. color(blue)(secx=1/cosx)`

`1`. To verify the given identity, start by working on the left side. Rewrite `tanx` in terms of `sinx` and `cosx`.

Left side:

`color(red)tanxsinx+cosx`

`=color(red)(sinx/cosx)*sinx+cosx`

`2`. Simplify.

`=sin^2x/cosx+cosx`

`=color(darkorange)(sin^2x+cos^2x)/cosx`

`3`. Simplify the numerator.

`=color(darkorange)1/cosx`

`=color(green)(|bar(ul(color(white)(a/a)secxcolor(white)(a/a)|)))`