Question

Tan×+cosx÷1+sinx?

Answer 1

`tanx+cosx/(1+sinx)=secx=1/cosx`

Explanation:

I'm assuming you meant `cosx-:(1+sinx)`, as `cosx-:1` is trivial.

When we have `1+sinx` in the denominator, we want to amplify the fraction by `1-sinx` to get `1-sin^2x=cos^2x`.

`:. E(x)=tanx+cosx/(1+sinx)*color(red)((1-sinx)/(1-sinx))`

`E(x) = tanx+(cosx(1-sinx))/(1-sin^2x)=tanx+(cosx(1-sinx))/cos^2x`

Simplifying the `cosx`:

`E(x)=tanx+(1-sinx)/cosx`

We know, by definition, that `tanx=sinx/cosx`.

`:. E(x) = sinx/cosx+(1-sinx)/cosx=(sinx+1-sinx)/cosx=1/cosx`

`=>color(red)(E(x)=secx)`