Question

Question #1cfd6

Answer 1

We want to prove:

`2tanxcos^2(x/2)=sinx+tanx`

We'll modify only the left-hand side of this equation. Let's start by trying to find a way to rewrite `cos^2(x/2)` using other identities. Start with the cosine double angle formula:

`cos(2x)=2cos^2x-1`

Which is the same as saying:

`cosx=2cos^2(x/2)-1`

So:

`2cos^2(x/2)=cosx+1`

Then our original expression on the left can become:

`2tanxcos^2(x/2)=tanx[2cos^2(x/2)]=tanx(cosx+1)`

Expanding this, it becomes:

`=tanxcosx+tanx`

Since `tanx=sinx/cosx`:

`=sinx/cosxcosx+tanx=sinx+tanx`

Which is the right-hand side of the equation, so we've proved the identity.