Question #1cfd6

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Question #1cfd6

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Answer 1 · 2017-03-17T18:40:17

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.

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