Question

Show that `(2cos^2(x/2)tanx)/(tanx) = (tanx+cosxtanx)/(tanx)`?

Answer 1

Okay, let's consider dividing both sides by `tanx` to begin with:

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

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

Notice how we can use the identity `sin^2(u) + cos^2(u) = 1`, where `u = x/2`:

`2cos^2(x/2) - [sin^2(x/2) + cos^2(x/2)] = cosx`

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

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

Then, we can use the identity `cos(u + v) = cosucosv - sinusinv`, where `u = v = x/2` and work backwards:

`cos(x/2)cos(x/2) - sin(x/2)sin(x/2) = cosx`

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

`color(blue)(cosx = cosx)`